# Linear Algebra/Basis/Solutions

## Solutions

This exercise is recommended for all readers.
Problem 1

Decide if each is a basis for ${\displaystyle \mathbb {R} ^{3}}$ .

1. ${\displaystyle \langle {\begin{pmatrix}1\\2\\3\end{pmatrix}},{\begin{pmatrix}3\\2\\1\end{pmatrix}},{\begin{pmatrix}0\\0\\1\end{pmatrix}}\rangle }$
2. ${\displaystyle \langle {\begin{pmatrix}1\\2\\3\end{pmatrix}},{\begin{pmatrix}3\\2\\1\end{pmatrix}}\rangle }$
3. ${\displaystyle \langle {\begin{pmatrix}0\\2\\-1\end{pmatrix}},{\begin{pmatrix}1\\1\\1\end{pmatrix}},{\begin{pmatrix}2\\5\\0\end{pmatrix}}\rangle }$
4. ${\displaystyle \langle {\begin{pmatrix}0\\2\\-1\end{pmatrix}},{\begin{pmatrix}1\\1\\1\end{pmatrix}},{\begin{pmatrix}1\\3\\0\end{pmatrix}}\rangle }$
Answer

By Theorem 1.12, each is a basis if and only if each vector in the space can be given in a unique way as a linear combination of the given vectors.

1. Yes this is a basis. The relation
${\displaystyle c_{1}{\begin{pmatrix}1\\2\\3\end{pmatrix}}+c_{2}{\begin{pmatrix}3\\2\\1\end{pmatrix}}+c_{3}{\begin{pmatrix}0\\0\\1\end{pmatrix}}={\begin{pmatrix}x\\y\\z\end{pmatrix}}}$
gives
${\displaystyle \left({\begin{array}{*{3}{c}|c}1&3&0&x\\2&2&0&y\\3&1&1&z\end{array}}\right){\xrightarrow[{-3\rho _{1}+\rho _{3}}]{-2\rho _{1}+\rho _{2}}}\;{\xrightarrow[{}]{2\rho _{2}+\rho _{3}}}\left({\begin{array}{*{3}{c}|c}1&3&0&x\\0&-4&0&-2x+y\\0&0&1&x-2y+z\end{array}}\right)}$
which has the unique solution ${\displaystyle c_{3}=x-2y+z}$ , ${\displaystyle c_{2}=x/2-y/4}$ , and ${\displaystyle c_{1}=-x/2+3y/4}$ .
2. This is not a basis. Setting it up as in the prior item
${\displaystyle c_{1}{\begin{pmatrix}1\\2\\3\end{pmatrix}}+c_{2}{\begin{pmatrix}3\\2\\1\end{pmatrix}}={\begin{pmatrix}x\\y\\z\end{pmatrix}}}$
gives a linear system whose solution
${\displaystyle \left({\begin{array}{*{2}{c}|c}1&3&x\\2&2&y\\3&1&z\end{array}}\right){\xrightarrow[{-3\rho _{1}+\rho _{3}}]{-2\rho _{1}+\rho _{2}}}\;{\xrightarrow[{}]{2\rho _{2}+\rho _{3}}}\left({\begin{array}{*{2}{c}|c}1&3&x\\0&-4&-2x+y\\0&0&x-2y+z\end{array}}\right)}$
is possible if and only if the three-tall vector's components ${\displaystyle x}$ , ${\displaystyle y}$ , and ${\displaystyle z}$  satisfy ${\displaystyle x-2y+z=0}$ . For instance, we can find the coefficients ${\displaystyle c_{1}}$  and ${\displaystyle c_{2}}$  that work when ${\displaystyle x=1}$ , ${\displaystyle y=1}$ , and ${\displaystyle z=1}$ . However, there are no ${\displaystyle c}$ 's that work for ${\displaystyle x=1}$ , ${\displaystyle y=1}$ , and ${\displaystyle z=2}$ . Thus this is not a basis; it does not span the space.
3. Yes, this is a basis. Setting up the relationship leads to this reduction
${\displaystyle \left({\begin{array}{*{3}{c}|c}0&1&2&x\\2&1&5&y\\-1&1&0&z\end{array}}\right){\xrightarrow[{}]{rho_{1}\leftrightarrow \rho _{3}}}\;{\xrightarrow[{}]{\rho _{1}+\rho _{2}}}\;{\xrightarrow[{}]{(1/3)\rho _{2}+\rho _{3}}}\left({\begin{array}{*{3}{c}|c}-1&1&0&z\\0&3&5&y+2z\\0&0&1/3&x-y/3-2z/3\end{array}}\right)}$
which has a unique solution for each triple of components ${\displaystyle x}$ , ${\displaystyle y}$ , and ${\displaystyle z}$ .
4. No, this is not a basis. The reduction
${\displaystyle \left({\begin{array}{*{3}{c}|c}0&1&1&x\\2&1&3&y\\-1&1&0&z\end{array}}\right){\xrightarrow[{}]{rho_{1}\leftrightarrow \rho _{3}}}\;{\xrightarrow[{}]{2\rho _{1}+\rho _{2}}}{\xrightarrow[{}]{-1/3)\rho _{2}+\rho _{3}}}\left({\begin{array}{*{3}{c}|c}-1&1&0&z\\0&3&3&y+2z\\0&0&0&x-y/3-2z/3\end{array}}\right)}$
which does not have a solution for each triple ${\displaystyle x}$ , ${\displaystyle y}$ , and ${\displaystyle z}$ . Instead, the span of the given set includes only those three-tall vectors where ${\displaystyle x=y/3+2z/3}$ .
This exercise is recommended for all readers.
Problem 2

Represent the vector with respect to the basis.

1. ${\displaystyle {\begin{pmatrix}1\\2\end{pmatrix}}}$ , ${\displaystyle B=\langle {\begin{pmatrix}1\\1\end{pmatrix}},{\begin{pmatrix}-1\\1\end{pmatrix}}\rangle \subseteq \mathbb {R} ^{2}}$
2. ${\displaystyle x^{2}+x^{3}}$ , ${\displaystyle D=\langle 1,1+x,1+x+x^{2},1+x+x^{2}+x^{3}\rangle \subseteq {\mathcal {P}}_{3}}$
3. ${\displaystyle {\begin{pmatrix}0\\-1\\0\\1\end{pmatrix}}}$ , ${\displaystyle {\mathcal {E}}_{4}\subseteq \mathbb {R} ^{4}}$
Answer
1. We solve
${\displaystyle c_{1}{\begin{pmatrix}1\\1\end{pmatrix}}+c_{2}{\begin{pmatrix}-1\\1\end{pmatrix}}={\begin{pmatrix}1\\2\end{pmatrix}}}$
with
${\displaystyle \left({\begin{array}{*{2}{c}|c}1&-1&1\\1&1&2\end{array}}\right){\xrightarrow[{}]{\rho _{1}+\rho _{2}}}\left({\begin{array}{*{2}{c}|c}1&-1&1\\0&2&1\end{array}}\right)}$
and conclude that ${\displaystyle c_{2}=1/2}$  and so ${\displaystyle c_{1}=3/2}$ . Thus, the representation is this.
${\displaystyle {\rm {Rep}}_{B}({\begin{pmatrix}1\\2\end{pmatrix}})={\begin{pmatrix}3/2\\1/2\end{pmatrix}}_{B}}$
2. The relationship ${\displaystyle c_{1}\cdot (1)+c_{2}\cdot (1+x)+c_{3}\cdot (1+x+x^{2})+c_{4}\cdot (1+x+x^{2}+x^{3})=x^{2}+x^{3}}$  is easily solved by eye to give that ${\displaystyle c_{4}=1}$ , ${\displaystyle c_{3}=0}$ , ${\displaystyle c_{2}=-1}$ , and ${\displaystyle c_{1}=0}$ .
${\displaystyle {\rm {Rep}}_{D}(x^{2}+x^{3})={\begin{pmatrix}0\\-1\\0\\1\end{pmatrix}}_{D}}$
3. ${\displaystyle {\rm {Rep}}_{{\mathcal {E}}_{4}}({\begin{pmatrix}0\\-1\\0\\1\end{pmatrix}})={\begin{pmatrix}0\\-1\\0\\1\end{pmatrix}}_{{\mathcal {E}}_{4}}}$
Problem 3

Find a basis for ${\displaystyle {\mathcal {P}}_{2}}$ , the space of all quadratic polynomials. Must any such basis contain a polynomial of each degree:~degree zero, degree one, and degree two?

Answer

One basis is ${\displaystyle \langle 1,x,x^{2}\rangle }$ . There are bases for ${\displaystyle {\mathcal {P}}_{2}}$  that do not contain any polynomials of degree one or degree zero. One is ${\displaystyle \langle 1+x+x^{2},x+x^{2},x^{2}\rangle }$ . (Every basis has at least one polynomial of degree two, though.)

Problem 4

Find a basis for the solution set of this system.

${\displaystyle {\begin{array}{*{4}{rc}r}x_{1}&-&4x_{2}&+&3x_{3}&-&x_{4}&=&0\\2x_{1}&-&8x_{2}&+&6x_{3}&-&2x_{4}&=&0\end{array}}}$
Answer

The reduction

${\displaystyle \left({\begin{array}{*{4}{c}|c}1&-4&3&-1&0\\2&-8&6&-2&0\end{array}}\right){\xrightarrow[{}]{2\rho _{1}+\rho _{2}}}\left({\begin{array}{*{4}{c}|c}1&-4&3&-1&0\\0&0&0&0&0\end{array}}\right)}$

gives that the only condition is that ${\displaystyle x_{1}=4x_{2}-3x_{3}+x_{4}}$ . The solution set is

${\displaystyle \{{\begin{pmatrix}4x_{2}-3x_{3}+x_{4}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}\,{\big |}\,x_{2},x_{3},x_{4}\in \mathbb {R} \}=\{x_{2}{\begin{pmatrix}4\\1\\0\\0\end{pmatrix}}+x_{3}{\begin{pmatrix}-3\\0\\1\\0\end{pmatrix}}+x_{4}{\begin{pmatrix}1\\0\\0\\1\end{pmatrix}}\,{\big |}\,x_{2},x_{3},x_{4}\in \mathbb {R} \}}$

and so the obvious candidate for the basis is this.

${\displaystyle \langle {\begin{pmatrix}4\\1\\0\\0\end{pmatrix}},{\begin{pmatrix}-3\\0\\1\\0\end{pmatrix}},{\begin{pmatrix}1\\0\\0\\1\end{pmatrix}}\rangle }$

We've shown that this spans the space, and showing it is also linearly independent is routine.

This exercise is recommended for all readers.
Problem 5

Find a basis for ${\displaystyle {\mathcal {M}}_{2\!\times \!2}}$ , the space of ${\displaystyle 2\!\times \!2}$  matrices.

Answer

There are many bases. This is an easy one.

${\displaystyle \langle {\begin{pmatrix}1&0\\0&0\end{pmatrix}},{\begin{pmatrix}0&1\\0&0\end{pmatrix}},{\begin{pmatrix}0&0\\1&0\end{pmatrix}},{\begin{pmatrix}0&0\\0&1\end{pmatrix}}\rangle }$
This exercise is recommended for all readers.
Problem 6

Find a basis for each.

1. The subspace ${\displaystyle \{a_{2}x^{2}+a_{1}x+a_{0}\,{\big |}\,a_{2}-2a_{1}=a_{0}\}}$  of ${\displaystyle {\mathcal {P}}_{2}}$
2. The space of three-wide row vectors whose first and second components add to zero
3. This subspace of the ${\displaystyle 2\!\times \!2}$  matrices
${\displaystyle \{{\begin{pmatrix}a&b\\0&c\end{pmatrix}}\,{\big |}\,c-2b=0\}}$
Answer

For each item, many answers are possible.

1. One way to proceed is to parametrize by expressing the ${\displaystyle a_{2}}$  as a combination of the other two ${\displaystyle a_{2}=2a_{1}+a_{0}}$ . Then ${\displaystyle a_{2}x^{2}+a_{1}x+a_{0}}$  is ${\displaystyle (2a_{1}+a_{0})x^{2}+a_{1}x+a_{0}}$  and
${\displaystyle \{(2a_{1}+a_{0})x^{2}+a_{1}x+a_{0}\,{\big |}\,a_{1},a_{0}\in \mathbb {R} \}=\{a_{1}\cdot (2x^{2}+x)+a_{0}\cdot (x^{2}+1)\,{\big |}\,a_{1},a_{0}\in \mathbb {R} \}}$
suggests ${\displaystyle \langle 2x^{2}+x,x^{2}+1\rangle }$ . This only shows that it spans, but checking that it is linearly independent is routine.
2. Parametrize ${\displaystyle \{{\begin{pmatrix}a&b&c\end{pmatrix}}\,{\big |}\,a+b=0\}}$  to get ${\displaystyle \{{\begin{pmatrix}-b&b&c\end{pmatrix}}\,{\big |}\,b,c\in \mathbb {R} \}}$ , which suggests using the sequence ${\displaystyle \langle {\begin{pmatrix}-1&1&0\end{pmatrix}},{\begin{pmatrix}0&0&1\end{pmatrix}}\rangle }$ . We've shown that it spans, and checking that it is linearly independent is easy.
3. Rewriting
${\displaystyle \{{\begin{pmatrix}a&b\\0&2b\end{pmatrix}}\,{\big |}\,a,b\in \mathbb {R} \}=\{a\cdot {\begin{pmatrix}1&0\\0&0\end{pmatrix}}+b\cdot {\begin{pmatrix}0&1\\0&2\end{pmatrix}}\,{\big |}\,a,b\in \mathbb {R} \}}$
suggests this for the basis.
${\displaystyle \langle {\begin{pmatrix}1&0\\0&0\end{pmatrix}},{\begin{pmatrix}0&1\\0&2\end{pmatrix}}\rangle }$
Problem 7

Check Example 1.6.

Answer

We will show that the second is a basis; the first is similar. We will show this straight from the definition of a basis, because this example appears before Theorem 1.12.

To see that it is linearly independent, we set up ${\displaystyle c_{1}\cdot (\cos \theta -\sin \theta )+c_{2}\cdot (2\cos \theta +3\sin \theta )=0\cos \theta +0\sin \theta }$ . Taking ${\displaystyle \theta =0}$  and ${\displaystyle \theta =\pi /2}$  gives this system

${\displaystyle {\begin{array}{*{2}{rc}r}c_{1}\cdot 1&+&c_{2}\cdot 2&=&0\\c_{1}\cdot (-1)&+&c_{2}\cdot 3&=&0\end{array}}\;{\xrightarrow[{}]{\rho _{1}+\rho _{2}}}\;{\begin{array}{*{2}{rc}r}c_{1}&+&2c_{2}&=&0\\&+&5c_{2}&=&0\end{array}}}$

which shows that ${\displaystyle c_{1}=0}$  and ${\displaystyle c_{2}=0}$ .

The calculation for span is also easy; for any ${\displaystyle x,y\in \mathbb {R} }$ , we have that ${\displaystyle c_{1}\cdot (\cos \theta -\sin \theta )+c_{2}\cdot (2\cos \theta +3\sin \theta )=x\cos \theta +y\sin \theta }$  gives that ${\displaystyle c_{2}=x/5+y/5}$  and that ${\displaystyle c_{1}=3x/5-2y/5}$ , and so the span is the entire space.

This exercise is recommended for all readers.
Problem 8

Find the span of each set and then find a basis for that span.

1. ${\displaystyle \{1+x,1+2x\}}$  in ${\displaystyle {\mathcal {P}}_{2}}$
2. ${\displaystyle \{2-2x,3+4x^{2}\}}$  in ${\displaystyle {\mathcal {P}}_{2}}$
Answer
1. Asking which ${\displaystyle a_{0}+a_{1}x+a_{2}x^{2}}$  can be expressed as ${\displaystyle c_{1}\cdot (1+x)+c_{2}\cdot (1+2x)}$  gives rise to three linear equations, describing the coefficients of ${\displaystyle x^{2}}$ , ${\displaystyle x}$ , and the constants.
${\displaystyle {\begin{array}{*{2}{rc}r}c_{1}&+&c_{2}&=&a_{0}\\c_{1}&+&2c_{2}&=&a_{1}\\&&0&=&a_{2}\end{array}}}$
Gauss' method with back-substitution shows, provided that ${\displaystyle a_{2}=0}$ , that ${\displaystyle c_{2}=-a_{0}+a_{1}}$  and ${\displaystyle c_{1}=2a_{0}-a_{1}}$ . Thus, with ${\displaystyle a_{2}=0}$ , we can compute appropriate ${\displaystyle c_{1}}$  and ${\displaystyle c_{2}}$  for any ${\displaystyle a_{0}}$  and ${\displaystyle a_{1}}$ . So the span is the entire set of linear polynomials ${\displaystyle \{a_{0}+a_{1}x\,{\big |}\,a_{0},a_{1}\in \mathbb {R} \}}$ . Parametrizing that set ${\displaystyle \{a_{0}\cdot 1+a_{1}\cdot x\,{\big |}\,a_{0},a_{1}\in \mathbb {R} \}}$  suggests a basis ${\displaystyle \langle 1,x\rangle }$  (we've shown that it spans; checking linear independence is easy).
2. With
${\displaystyle a_{0}+a_{1}x+a_{2}x^{2}=c_{1}\cdot (2-2x)+c_{2}\cdot (3+4x^{2})=(2c_{1}+3c_{2})+(-2c_{1})x+(4c_{2})x^{2}}$
we get this system.
${\displaystyle {\begin{array}{*{2}{rc}r}2c_{1}&+&3c_{2}&=&a_{0}\\-2c_{1}&&&=&a_{1}\\&&4c_{2}&=&a_{2}\end{array}}\;{\xrightarrow[{}]{\rho _{1}+\rho _{2}}}\;{\xrightarrow[{}]{-4/3)\rho _{2}+\rho _{3}}}\;{\begin{array}{*{2}{rc}r}2c_{1}&+&3c_{2}&=&a_{0}\\&&3c_{2}&=&a_{0}+a_{1}\\&&0&=&(-4/3)a_{0}-(4/3)a_{1}+a_{2}\end{array}}}$
Thus, the only quadratic polynomials ${\displaystyle a_{0}+a_{1}x+a_{2}x^{2}}$  with associated ${\displaystyle c}$ 's are the ones such that ${\displaystyle 0=(-4/3)a_{0}-(4/3)a_{1}+a_{2}}$ . Hence the span is ${\displaystyle \{(-a_{1}+(3/4)a_{2})+a_{1}x+a_{2}x^{2}\,{\big |}\,a_{1},a_{2}\in \mathbb {R} \}}$ . Parametrizing gives ${\displaystyle \{a_{1}\cdot (-1+x)+a_{2}\cdot ((3/4)+x^{2})\,{\big |}\,a_{1},a_{2}\in \mathbb {R} \}}$ , which suggests ${\displaystyle \langle -1+x,(3/4)+x^{2}\rangle }$  (checking that it is linearly independent is routine).
This exercise is recommended for all readers.
Problem 9

Find a basis for each of these subspaces of the space ${\displaystyle {\mathcal {P}}_{3}}$  of cubic polynomials.

1. The subspace of cubic polynomials ${\displaystyle p(x)}$  such that ${\displaystyle p(7)=0}$
2. The subspace of polynomials ${\displaystyle p(x)}$  such that ${\displaystyle p(7)=0}$  and ${\displaystyle p(5)=0}$
3. The subspace of polynomials ${\displaystyle p(x)}$  such that ${\displaystyle p(7)=0}$ , ${\displaystyle p(5)=0}$ , and~${\displaystyle p(3)=0}$
4. The space of polynomials ${\displaystyle p(x)}$  such that ${\displaystyle p(7)=0}$ , ${\displaystyle p(5)=0}$ , ${\displaystyle p(3)=0}$ , and~${\displaystyle p(1)=0}$
Answer
1. The subspace is ${\displaystyle \{a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}\,{\big |}\,a_{0}+7a_{1}+49a_{2}+343a_{3}=0\}}$ . Rewriting ${\displaystyle a_{0}=-7a_{1}-49a_{2}-343a_{3}}$  gives ${\displaystyle \{(-7a_{1}-49a_{2}-343a_{3})+a_{1}x+a_{2}x^{2}+a_{3}x^{3}\,{\big |}\,a_{1},a_{2},a_{3}\in \mathbb {R} \}}$ , which, on breaking out the parameters, suggests ${\displaystyle \langle -7+x,-49+x^{2},-343+x^{3}\rangle }$  for the basis (it is easily verified).
2. The given subspace is the collection of cubics ${\displaystyle p(x)=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}}$  such that ${\displaystyle a_{0}+7a_{1}+49a_{2}+343a_{3}=0}$  and ${\displaystyle a_{0}+5a_{1}+25a_{2}+125a_{3}=0}$ . Gauss' method
${\displaystyle {\begin{array}{*{4}{rc}r}a_{0}&+&7a_{1}&+&49a_{2}&+&343a_{3}&=&0\\a_{0}&+&5a_{1}&+&25a_{2}&+&125a_{3}&=&0\end{array}}\;{\xrightarrow[{}]{\rho _{1}+\rho _{2}}}\;{\begin{array}{*{4}{rc}r}a_{0}&+&7a_{1}&+&49a_{2}&+&343a_{3}&=&0\\&&-2a_{1}&-&24a_{2}&-&218a_{3}&=&0\end{array}}}$
gives that ${\displaystyle a_{1}=-12a_{2}-109a_{3}}$  and that ${\displaystyle a_{0}=35a_{2}+420a_{3}}$ . Rewriting ${\displaystyle (35a_{2}+420a_{3})+(-12a_{2}-109a_{3})x+a_{2}x^{2}+a_{3}x^{3}}$  as ${\displaystyle a_{2}\cdot (35-12x+x^{2})+a_{3}\cdot (420-109x+x^{3})}$  suggests this for a basis ${\displaystyle \langle 35-12x+x^{2},420-109x+x^{3}\rangle }$ . The above shows that it spans the space. Checking it is linearly independent is routine. (Comment. A worthwhile check is to verify that both polynomials in the basis have both seven and five as roots.)
3. Here there are three conditions on the cubics, that ${\displaystyle a_{0}+7a_{1}+49a_{2}+343a_{3}=0}$ , that ${\displaystyle a_{0}+5a_{1}+25a_{2}+125a_{3}=0}$ ,and that ${\displaystyle a_{0}+3a_{1}+9a_{2}+27a_{3}=0}$ . Gauss' method
${\displaystyle {\begin{array}{*{4}{rc}r}a_{0}&+&7a_{1}&+&49a_{2}&+&343a_{3}&=&0\\a_{0}&+&5a_{1}&+&25a_{2}&+&125a_{3}&=&0\\a_{0}&+&3a_{1}&+&9a_{2}&+&27a_{3}&=&0\end{array}}\;{\xrightarrow[{-\rho _{1}+\rho _{3}}]{-\rho _{1}+\rho _{2}}}\;{\xrightarrow[{}]{2\rho _{2}+\rho _{3}}}\;{\begin{array}{*{4}{rc}r}a_{0}&+&7a_{1}&+&49a_{2}&+&343a_{3}&=&0\\&&-2a_{1}&-&24a_{2}&-&218a_{3}&=&0\\&&&&8a_{2}&+&120a_{3}&=&0\end{array}}}$
yields the single free variable ${\displaystyle a_{3}}$ , with ${\displaystyle a_{2}=-15a_{3}}$ , ${\displaystyle a_{1}=71a_{3}}$ , and ${\displaystyle a_{0}=-105a_{3}}$ . The parametrization is this.
${\displaystyle \{(-105a_{3})+(71a_{3})x+(-15a_{3})x^{2}+(a_{3})x^{3}\,{\big |}\,a_{3}\in \mathbb {R} \}=\{a_{3}\cdot (-105+71x-15x^{2}+x^{3})\,{\big |}\,a_{3}\in \mathbb {R} \}}$
Therefore, a good candidate for the basis is ${\displaystyle \langle -105+71x-15x^{2}+x^{3}\rangle }$ . It spans the space by the work above. It is clearly linearly independent because it is a one-element set (with that single element not the zero object of the space). Thus, any cubic through the three points ${\displaystyle (7,0)}$ , ${\displaystyle (5,0)}$ , and ${\displaystyle (3,0)}$  is a multiple of this one. (Comment. As in the prior question, a worthwhile check is to verify that plugging seven, five, and three into this polynomial yields zero each time.)
4. This is the trivial subspace of ${\displaystyle {\mathcal {P}}_{3}}$ . Thus, the basis is empty ${\displaystyle \langle \rangle }$ .

Remark. The polynomial in the third item could alternatively have been derived by multiplying out ${\displaystyle (x-7)(x-5)(x-3)}$ .

Problem 10

We've seen that it is possible for a basis to remain a basis when it is reordered. Must it always remain a basis?

Answer

Yes. Linear independence and span are unchanged by reordering.

Problem 11

Can a basis contain a zero vector?

Answer

No linearly independent set contains a zero vector.

This exercise is recommended for all readers.
Problem 12

Let ${\displaystyle \langle {\vec {\beta }}_{1},{\vec {\beta }}_{2},{\vec {\beta }}_{3}\rangle }$  be a basis for a vector space.

1. Show that ${\displaystyle \langle c_{1}{\vec {\beta }}_{1},c_{2}{\vec {\beta }}_{2},c_{3}{\vec {\beta }}_{3}\rangle }$  is a basis when ${\displaystyle c_{1},c_{2},c_{3}\neq 0}$ . What happens when at least one ${\displaystyle c_{i}}$  is ${\displaystyle 0}$ ?
2. Prove that ${\displaystyle \langle {\vec {\alpha }}_{1},{\vec {\alpha }}_{2},{\vec {\alpha }}_{3}\rangle }$  is a basis where ${\displaystyle {\vec {\alpha }}_{i}={\vec {\beta }}_{1}+{\vec {\beta }}_{i}}$ .
Answer
1. To show that it is linearly independent, note that ${\displaystyle d_{1}(c_{1}{\vec {\beta }}_{1})+d_{2}(c_{2}{\vec {\beta }}_{2})+d_{3}(c_{3}{\vec {\beta }}_{3})={\vec {0}}}$  gives that ${\displaystyle (d_{1}c_{1}){\vec {\beta }}_{1}+(d_{2}c_{2}){\vec {\beta }}_{2}+(d_{3}c_{3}){\vec {\beta }}_{3}={\vec {0}}}$ , which in turn implies that each ${\displaystyle d_{i}c_{i}}$  is zero. But with ${\displaystyle c_{i}\neq 0}$  that means that each ${\displaystyle d_{i}}$  is zero. Showing that it spans the space is much the same; because ${\displaystyle \langle {\vec {\beta }}_{1},{\vec {\beta }}_{2},{\vec {\beta }}_{3}\rangle }$  is a basis, and so spans the space, we can for any ${\displaystyle {\vec {v}}}$  write ${\displaystyle {\vec {v}}=d_{1}{\vec {\beta }}_{1}+d_{2}{\vec {\beta }}_{2}+d_{3}{\vec {\beta }}_{3}}$ , and then ${\displaystyle {\vec {v}}=(d_{1}/c_{1})(c_{1}{\vec {\beta }}_{1})+(d_{2}/c_{2})(c_{2}{\vec {\beta }}_{2})+(d_{3}/c_{3})(c_{3}{\vec {\beta }}_{3})}$ . If any of the scalars are zero then the result is not a basis, because it is not linearly independent.
2. Showing that ${\displaystyle \langle 2{\vec {\beta }}_{1},{\vec {\beta }}_{1}+{\vec {\beta }}_{2},{\vec {\beta }}_{1}+{\vec {\beta }}_{3}\rangle }$  is linearly independent is easy. To show that it spans the space, assume that ${\displaystyle {\vec {v}}=d_{1}{\vec {\beta }}_{1}+d_{2}{\vec {\beta }}_{2}+d_{3}{\vec {\beta }}_{3}}$ . Then, we can represent the same ${\displaystyle {\vec {v}}}$  with respect to ${\displaystyle \langle 2{\vec {\beta }}_{1},{\vec {\beta }}_{1}+{\vec {\beta }}_{2},{\vec {\beta }}_{1}+{\vec {\beta }}_{3}\rangle }$  in this way ${\displaystyle {\vec {v}}=(1/2)(d_{1}-d_{2}-d_{3})(2{\vec {\beta }}_{1})+d_{2}({\vec {\beta }}_{1}+{\vec {\beta }}_{2})+d_{3}({\vec {\beta }}_{1}+{\vec {\beta }}_{3})}$ .
Problem 13

Find one vector ${\displaystyle {\vec {v}}}$  that will make each into a basis for the space.

1. ${\displaystyle \langle {\begin{pmatrix}1\\1\end{pmatrix}},{\vec {v}}\rangle }$  in ${\displaystyle \mathbb {R} ^{2}}$
2. ${\displaystyle \langle {\begin{pmatrix}1\\1\\0\end{pmatrix}},{\begin{pmatrix}0\\1\\0\end{pmatrix}},{\vec {v}}\rangle }$  in ${\displaystyle \mathbb {R} ^{3}}$
3. ${\displaystyle \langle x,1+x^{2},{\vec {v}}\rangle }$  in ${\displaystyle {\mathcal {P}}_{2}}$
Answer

Each forms a linearly independent set if ${\displaystyle {\vec {v}}}$  is omitted. To preserve linear independence, we must expand the span of each. That is, we must determine the span of each (leaving ${\displaystyle {\vec {v}}}$  out), and then pick a ${\displaystyle {\vec {v}}}$  lying outside of that span. Then to finish, we must check that the result spans the entire given space. Those checks are routine.

1. Any vector that is not a multiple of the given one, that is, any vector that is not on the line ${\displaystyle y=x}$  will do here. One is ${\displaystyle {\vec {v}}={\vec {e}}_{1}}$ .
2. By inspection, we notice that the vector ${\displaystyle {\vec {e}}_{3}}$  is not in the span of the set of the two given vectors. The check that the resulting set is a basis for ${\displaystyle \mathbb {R} ^{3}}$  is routine.
3. For any member of the span ${\displaystyle \{c_{1}\cdot (x)+c_{2}\cdot (1+x^{2})\,{\big |}\,c_{1},c_{2}\in \mathbb {R} \}}$ , the coefficient of ${\displaystyle x^{2}}$  equals the constant term. So we expand the span if we add a quadratic without this property, say, ${\displaystyle {\vec {v}}=1-x^{2}}$ . The check that the result is a basis for ${\displaystyle {\mathcal {P}}_{2}}$  is easy.
This exercise is recommended for all readers.
Problem 14

Where ${\displaystyle \langle {\vec {\beta }}_{1},\dots ,{\vec {\beta }}_{n}\rangle }$  is a basis, show that in this equation

${\displaystyle c_{1}{\vec {\beta }}_{1}+\dots +c_{k}{\vec {\beta }}_{k}=c_{k+1}{\vec {\beta }}_{k+1}+\dots +c_{n}{\vec {\beta }}_{n}}$

each of the ${\displaystyle c_{i}}$ 's is zero. Generalize.

Answer

To show that each scalar is zero, simply subtract ${\displaystyle c_{1}{\vec {\beta }}_{1}+\dots +c_{k}{\vec {\beta }}_{k}-c_{k+1}{\vec {\beta }}_{k+1}-\dots -c_{n}{\vec {\beta }}_{n}={\vec {0}}}$ . The obvious generalization is that in any equation involving only the ${\displaystyle {\vec {\beta }}}$ 's, and in which each ${\displaystyle {\vec {\beta }}}$  appears only once, each scalar is zero. For instance, an equation with a combination of the even-indexed basis vectors (i.e., ${\displaystyle {\vec {\beta }}_{2}}$ , ${\displaystyle {\vec {\beta }}_{4}}$ , etc.) on the right and the odd-indexed basis vectors on the left also gives the conclusion that all of the coefficients are zero.

Problem 15

A basis contains some of the vectors from a vector space; can it contain them all?

Answer

No; no linearly independent set contains the zero vector.

Problem 16

Theorem 1.12 shows that, with respect to a basis, every linear combination is unique. If a subset is not a basis, can linear combinations be not unique? If so, must they be?

Answer

Here is a subset of ${\displaystyle \mathbb {R} ^{2}}$  that is not a basis, and two different linear combinations of its elements that sum to the same vector.

${\displaystyle \{{\begin{pmatrix}1\\2\end{pmatrix}},{\begin{pmatrix}2\\4\end{pmatrix}}\}\qquad 2\cdot {\begin{pmatrix}1\\2\end{pmatrix}}+0\cdot {\begin{pmatrix}2\\4\end{pmatrix}}=0\cdot {\begin{pmatrix}1\\2\end{pmatrix}}+1\cdot {\begin{pmatrix}2\\4\end{pmatrix}}}$

Thus, when a subset is not a basis, it can be the case that its linear combinations are not unique.

But just because a subset is not a basis does not imply that its combinations must be not unique. For instance, this set

${\displaystyle \{{\begin{pmatrix}1\\2\end{pmatrix}}\}}$

does have the property that

${\displaystyle c_{1}\cdot {\begin{pmatrix}1\\2\end{pmatrix}}=c_{2}\cdot {\begin{pmatrix}1\\2\end{pmatrix}}}$

implies that ${\displaystyle c_{1}=c_{2}}$ . The idea here is that this subset fails to be a basis because it fails to span the space; the proof of the theorem establishes that linear combinations are unique if and only if the subset is linearly independent.

This exercise is recommended for all readers.
Problem 17

A square matrix is symmetric if for all indices ${\displaystyle i}$  and ${\displaystyle j}$ , entry ${\displaystyle i,j}$  equals entry ${\displaystyle j,i}$ .

1. Find a basis for the vector space of symmetric ${\displaystyle 2\!\times \!2}$  matrices.
2. Find a basis for the space of symmetric ${\displaystyle 3\!\times \!3}$  matrices.
3. Find a basis for the space of symmetric ${\displaystyle n\!\times \!n}$  matrices.
Answer
1. Describing the vector space as
${\displaystyle \{{\begin{pmatrix}a&b\\b&c\end{pmatrix}}\,{\big |}\,a,b,c\in \mathbb {R} \}}$
suggests this for a basis.
${\displaystyle \langle {\begin{pmatrix}1&0\\0&0\end{pmatrix}},{\begin{pmatrix}0&0\\0&1\end{pmatrix}},{\begin{pmatrix}0&1\\1&0\end{pmatrix}}\rangle }$
Verification is easy.
2. This is one possible basis.
${\displaystyle \langle {\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}},{\begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix}},{\begin{pmatrix}0&0&0\\0&0&0\\0&0&1\end{pmatrix}},{\begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix}},{\begin{pmatrix}0&0&1\\0&0&0\\1&0&0\end{pmatrix}},{\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix}}\rangle }$
3. As in the prior two questions, we can form a basis from two kinds of matrices. First are the matrices with a single one on the diagonal and all other entries zero (there are ${\displaystyle n}$  of those matrices). Second are the matrices with two opposed off-diagonal entries are ones and all other entries are zeros. (That is, all entries in ${\displaystyle M}$  are zero except that ${\displaystyle m_{i,j}}$  and ${\displaystyle m_{j,i}}$  are one.)
This exercise is recommended for all readers.
Problem 18

We can show that every basis for ${\displaystyle \mathbb {R} ^{3}}$  contains the same number of vectors.

1. Show that no linearly independent subset of ${\displaystyle \mathbb {R} ^{3}}$  contains more than three vectors.
2. Show that no spanning subset of ${\displaystyle \mathbb {R} ^{3}}$  contains fewer than three vectors. (Hint. Recall how to calculate the span of a set and show that this method, when applied to two vectors, cannot yield all of ${\displaystyle \mathbb {R} ^{3}}$ .)
Answer
1. Any four vectors from ${\displaystyle \mathbb {R} ^{3}}$  are linearly related because the vector equation
${\displaystyle c_{1}{\begin{pmatrix}x_{1}\\y_{1}\\z_{1}\end{pmatrix}}+c_{2}{\begin{pmatrix}x_{2}\\y_{2}\\z_{2}\end{pmatrix}}+c_{3}{\begin{pmatrix}x_{3}\\y_{3}\\z_{3}\end{pmatrix}}+c_{4}{\begin{pmatrix}x_{4}\\y_{4}\\z_{4}\end{pmatrix}}={\begin{pmatrix}0\\0\\0\end{pmatrix}}}$
gives rise to a linear system
${\displaystyle {\begin{array}{*{4}{rc}r}x_{1}c_{1}&+&x_{2}c_{2}&+&x_{3}c_{3}&+&x_{4}c_{4}&=&0\\y_{1}c_{1}&+&y_{2}c_{2}&+&y_{3}c_{3}&+&y_{4}c_{4}&=&0\\z_{1}c_{1}&+&z_{2}c_{2}&+&z_{3}c_{3}&+&z_{4}c_{4}&=&0\end{array}}}$
that is homogeneous (and so has a solution) and has four unknowns but only three equations, and therefore has nontrivial solutions. (Of course, this argument applies to any subset of ${\displaystyle \mathbb {R} ^{3}}$  with four or more vectors.)
2. Given ${\displaystyle x_{1}}$ , ..., ${\displaystyle z_{2}}$ ,
${\displaystyle S=\{{\begin{pmatrix}x_{1}\\y_{1}\\z_{1}\end{pmatrix}},{\begin{pmatrix}x_{2}\\y_{2}\\z_{2}\end{pmatrix}}\}}$
to decide which vectors
${\displaystyle {\begin{pmatrix}x\\y\\z\end{pmatrix}}}$
are in the span of ${\displaystyle S}$ , set up
${\displaystyle c_{1}{\begin{pmatrix}x_{1}\\y_{1}\\z_{1}\end{pmatrix}}+c_{2}{\begin{pmatrix}x_{2}\\y_{2}\\z_{2}\end{pmatrix}}={\begin{pmatrix}x\\y\\z\end{pmatrix}}}$
and row reduce the resulting system.
${\displaystyle {\begin{array}{*{2}{rc}r}x_{1}c_{1}&+&x_{2}c_{2}&=&x\\y_{1}c_{1}&+&y_{2}c_{2}&=&y\\z_{1}c_{1}&+&z_{2}c_{2}&=&z\end{array}}}$
There are two variables ${\displaystyle c_{1}}$  and ${\displaystyle c_{2}}$  but three equations, so when Gauss' method finishes, on the bottomrow there will be some relationship of the form ${\displaystyle 0=m_{1}x+m_{2}y+m_{3}z}$ . Hence, vectors in the span of the two-element set ${\displaystyle S}$  must satisfy some restriction. Hence the span is not all of ${\displaystyle \mathbb {R} ^{3}}$ .
Problem 19

One of the exercises in the Subspaces subsection shows that the set

${\displaystyle \{{\begin{pmatrix}x\\y\\z\end{pmatrix}}\,{\big |}\,x+y+z=1\}}$

is a vector space under these operations.

${\displaystyle {\begin{pmatrix}x_{1}\\y_{1}\\z_{1}\end{pmatrix}}+{\begin{pmatrix}x_{2}\\y_{2}\\z_{2}\end{pmatrix}}={\begin{pmatrix}x_{1}+x_{2}-1\\y_{1}+y_{2}\\z_{1}+z_{2}\end{pmatrix}}\qquad r{\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}rx-r+1\\ry\\rz\end{pmatrix}}}$

Find a basis.

Answer

We have (using these peculiar operations with care)

${\displaystyle \{{\begin{pmatrix}1-y-z\\y\\z\end{pmatrix}}\,{\big |}\,y,z\in \mathbb {R} \}=\{{\begin{pmatrix}-y+1\\y\\0\end{pmatrix}}+{\begin{pmatrix}-z+1\\0\\z\end{pmatrix}}\,{\big |}\,y,z\in \mathbb {R} \}=\{y\cdot {\begin{pmatrix}0\\1\\0\end{pmatrix}}+z\cdot {\begin{pmatrix}0\\0\\1\end{pmatrix}}\,{\big |}\,y,z\in \mathbb {R} \}}$

and so a good candidate for a basis is this.

${\displaystyle \langle {\begin{pmatrix}0\\1\\0\end{pmatrix}},{\begin{pmatrix}0\\0\\1\end{pmatrix}}\rangle }$

To check linear independence we set up

${\displaystyle c_{1}{\begin{pmatrix}0\\1\\0\end{pmatrix}}+c_{2}{\begin{pmatrix}0\\0\\1\end{pmatrix}}={\begin{pmatrix}1\\0\\0\end{pmatrix}}}$

(the vector on the right is the zero object in this space). That yields the linear system

${\displaystyle {\begin{array}{*{3}{rc}r}(-c_{1}+1)&+&(-c_{2}+1)&-&1&=&1\\c_{1}&&&&&=&0\\&&c_{2}&&&=&0\end{array}}}$

with only the solution ${\displaystyle c_{1}=0}$  and ${\displaystyle c_{2}=0}$ . Checking the span is similar.