Linear Algebra/Representing Linear Maps with Matrices/Solutions

SolutionsEdit

This exercise is recommended for all readers.
Problem 1

Multiply the matrix

 

by each vector (or state "not defined").

  1.  
  2.  
  3.  
Answer
  1.  
  2. Not defined.
  3.  
Problem 2

Perform, if possible, each matrix-vector multiplication.

  1.  
  2.  
  3.  
Answer
  1.  
  2.  
  3. Not defined.
This exercise is recommended for all readers.
Problem 3

Solve this matrix equation.

 
Answer

Matrix-vector multiplication gives rise to a linear system.

 

Gaussian reduction shows that  ,  , and  .

This exercise is recommended for all readers.
Problem 4

For a homomorphism from   to   that sends

 

where does   go?

Answer

Here are two ways to get the answer.

First, obviously  , and so we can apply the general property of preservation of combinations to get  .

The other way uses the computation scheme developed in this subsection. Because we know where these elements of the space go, we consider this basis   for the domain. Arbitrarily, we can take   as a basis for the codomain. With those choices, we have that

 

and, as

 

the matrix-vector multiplication calculation gives this.

 

Thus,  , as above.

This exercise is recommended for all readers.
Problem 5

Assume that   is determined by this action.

 

Using the standard bases, find

  1. the matrix representing this map;
  2. a general formula for  .
Answer

Again, as recalled in the subsection, with respect to  , a column vector represents itself.

  1. To represent   with respect to   we take the images of the basis vectors from the domain, and represent them with respect to the basis for the codomain.
     
    These are adjoined to make the matrix.
     
  2. For any   in the domain  ,
     
    and so
     
    is the desired representation.
This exercise is recommended for all readers.
Problem 6

Let   be the derivative transformation.

  1. Represent   with respect to   where  .
  2. Represent   with respect to   where  .
Answer
  1. We must first find the image of each vector from the domain's basis, and then represent that image with respect to the codomain's basis.
     
    Those representations are then adjoined to make the matrix representing the map.
     
  2. Proceeding as in the prior item, we represent the images of the domain's basis vectors
     
    and adjoin to make the matrix.
     
This exercise is recommended for all readers.
Problem 7

Represent each linear map with respect to each pair of bases.

  1.   with respect to   where  , given by
     
  2.   with respect to   where  , given by
     
  3.   with respect to   where   and  , given by
     
  4.   with respect to   where   and  , given by
     
  5.   with respect to   where  , given by
     
Answer

For each, we must find the image of each of the domain's basis vectors, represent each image with respect to the codomain's basis, and then adjoin those representations to get the matrix.

  1. The basis vectors from the domain have these images
     
    and these images are represented with respect to the codomain's basis in this way.
     
    The matrix
     
    has   rows and columns.
  2. Once the images under this map of the domain's basis vectors are determined
     
    then they can be represented with respect to the codomain's basis
     
    and put together to make the matrix.
     
  3. The images of the basis vectors of the domain are
     
    and they are represented with respect to the codomain's basis as
     
    so the matrix is
     
    (this is an   matrix).
  4. Here, the images of the domain's basis vectors are
     
    and they are represented in the codomain as
     
    and so the matrix is this.
     
  5. The images of the basis vectors from the domain are
     
    which are represented as
     
    The resulting matrix
     
    is Pascal's triangle (recall that   is the number of ways to choose   things, without order and without repetition, from a set of size  ).
Problem 8

Represent the identity map on any nontrivial space with respect to  , where   is any basis.

Answer

Where the space is  -dimensional,

 

is the   identity matrix.

Problem 9

Represent, with respect to the natural basis, the transpose transformation on the space   of   matrices.

Answer

Taking this as the natural basis

 

the transpose map acts in this way

 

so that representing the images with respect to the codomain's basis and adjoining those column vectors together gives this.

 
Problem 10

Assume that   is a basis for a vector space. Represent with respect to   the transformation that is determined by each.

  1.  ,  ,  ,  
  2.  ,  ,  ,  
  3.  ,  ,  ,  
Answer
  1. With respect to the basis of the codomain, the images of the members of the basis of the domain are represented as
     
    and consequently, the matrix representing the transformation is this.
     
  2.  
  3.  
Problem 11

Example 1.8 shows how to represent the rotation transformation of the plane with respect to the standard basis. Express these other transformations also with respect to the standard basis.

  1. the dilation map  , which multiplies all vectors by the same scalar  
  2. the reflection map  , which reflects all all vectors across a line   through the origin
Answer
  1. The picture of   is this.

     

    This map's effect on the vectors in the standard basis for the domain is

     

    and those images are represented with respect to the codomain's basis (again, the standard basis) by themselves.

     

    Thus the representation of the dilation map is this.

     
  2. The picture of   is this.

     

    Some calculation (see Problem I.1.20) shows that when the line has slope  

     

    (the case of a line with undefined slope is separate but easy) and so the matrix representing reflection is this.

     
This exercise is recommended for all readers.
Problem 12

Consider a linear transformation of   determined by these two.

 
  1. Represent this transformation with respect to the standard bases.
  2. Where does the transformation send this vector?
     
  3. Represent this transformation with respect to these bases.
     
  4. Using   from the prior item, represent the transformation with respect to  .
Answer

Call the map  .

  1. To represent this map with respect to the standard bases, we must find, and then represent, the images of the vectors   and   from the domain's basis. The image of   is given. One way to find the image of   is by eye— we can see this.
     
    A more systemmatic way to find the image of   is to use the given information to represent the transformation, and then use that representation to determine the image. Taking this for a basis,
     
    the given information says this.
     
    As
     
    we have that
     
    and consequently we know that   (since, with respect to the standard basis, this vector is represented by itself). Therefore, this is the representation of   with respect to  .
     
  2. To use the matrix developed in the prior item, note that
     
    and so we have this is the representation, with respect to the codomain's basis, of the image of the given vector.
     
    Because the codomain's basis is the standard one, and so vectors in the codomain are represented by themselves, we have this.
     
  3. We first find the image of each member of  , and then represent those images with respect to  . For the first step, we can use the matrix developed earlier.
     
    Actually, for the second member of   there is no need to apply the matrix because the problem statement gives its image.
     
    Now representing those images with respect to   is routine.
     
    Thus, the matrix is this.
     
  4. We know the images of the members of the domain's basis from the prior item.
     
    We can compute the representation of those images with respect to the codomain's basis.
     
    Thus this is the matrix.
     
Problem 13

Suppose that   is nonsingular so that by Theorem II.2.21, for any basis   the image   is a basis for  .

  1. Represent the map   with respect to  .
  2. For a member   of the domain, where the representation of   has components  , ...,  , represent the image vector   with respect to the image basis  .
Answer
  1. The images of the members of the domain's basis are
     
    and those images are represented with respect to the codomain's basis in this way.
     
    Hence, the matrix is the identity.
     
  2. Using the matrix in the prior item, the representation is this.
     
Problem 14

Give a formula for the product of a matrix and  , the column vector that is all zeroes except for a single one in the  -th position.

Answer

The product

 

gives the  -th column of the matrix.

This exercise is recommended for all readers.
Problem 15

For each vector space of functions of one real variable, represent the derivative transformation with respect to  .

  1.  ,  
  2.  ,  
  3.  ,  
Answer
  1. The images of the basis vectors for the domain are   and  . Representing those with respect to the codomain's basis (again,  ) and adjoining the representations gives this matrix.
     
  2. The images of the vectors in the domain's basis are   and  . Representing with respect to the codomain's basis and adjoining gives this matrix.
     
  3. The images of the members of the domain's basis are  ,  ,  , and  . Representing these images with respect to   and adjoining gives this matrix.
     
Problem 16

Find the range of the linear transformation of   represented with respect to the standard bases by each matrix.

  1.  
  2.  
  3. a matrix of the form  
Answer
  1. It is the set of vectors of the codomain represented with respect to the codomain's basis in this way.
     
    As the codomain's basis is  , and so each vector is represented by itself, the range of this transformation is the  -axis.
  2. It is the set of vectors of the codomain represented in this way.
     
    With respect to   vectors represent themselves so this range is the   axis.
  3. The set of vectors represented with respect to   as
     
    is the line  , provided either   or   is not zero, and is the set consisting of just the origin if both are zero.
This exercise is recommended for all readers.
Problem 17

Can one matrix represent two different linear maps? That is, can  ?

Answer

Yes, for two reasons.

First, the two maps   and   need not have the same domain and codomain. For instance,

 

represents a map   with respect to the standard bases that sends

 

and also represents a   with respect to   and   that acts in this way.

 

The second reason is that, even if the domain and codomain of   and   coincide, different bases produce different maps. An example is the   identity matrix

 

which represents the identity map on   with respect to  . However, with respect to   for the domain but the basis   for the codomain, the same matrix   represents the map that swaps the first and second components

 

(that is, reflection about the line  ).

Problem 18

Prove Theorem 1.4.

Answer

We mimic Example 1.1, just replacing the numbers with letters.

Write   as   and   as  . By definition of representation of a map with respect to bases, the assumption that

 

means that  . And, by the definition of the representation of a vector with respect to a basis, the assumption that

 

means that  . Substituting gives

 

and so   is represented as required.

This exercise is recommended for all readers.
Problem 19

Example 1.8 shows how to represent rotation of all vectors in the plane through an angle   about the origin, with respect to the standard bases.

  1. Rotation of all vectors in three-space through an angle   about the  -axis is a transformation of  . Represent it with respect to the standard bases. Arrange the rotation so that to someone whose feet are at the origin and whose head is at  , the movement appears clockwise.
  2. Repeat the prior item, only rotate about the  -axis instead. (Put the person's head at  .)
  3. Repeat, about the  -axis.
  4. Extend the prior item to  . (Hint: "rotate about the  -axis" can be restated as "rotate parallel to the  -plane".)
Answer
  1. The picture is this.

     

    The images of the vectors from the domain's basis

     

    are represented with respect to the codomain's basis (again,  ) by themselves, so adjoining the representations to make the matrix gives this.

     
  2. The picture is similar to the one in the prior answer. The images of the vectors from the domain's basis
     
    are represented with respect to the codomain's basis   by themselves, so this is the matrix.
     
  3. To a person standing up, with the vertical  -axis, a rotation of the  -plane that is clockwise proceeds from the positive  -axis to the positive  -axis. That is, it rotates opposite to the direction in Example 1.8. The images of the vectors from the domain's basis
     
    are represented with respect to   by themselves, so the matrix is this.
     
  4.  
Problem 20 (Schur's Triangularization Lemma)
  1. Let   be a subspace of   and fix bases  . What is the relationship between the representation of a vector from   with respect to   and the representation of that vector (viewed as a member of  ) with respect to  ?
  2. What about maps?
  3. Fix a basis   for   and observe that the spans
     
    form a strictly increasing chain of subspaces. Show that for any linear map   there is a chain   of subspaces of   such that
     
    for each  .
  4. Conclude that for every linear map   there are bases   so the matrix representing   with respect to   is upper-triangular (that is, each entry   with   is zero).
  5. Is an upper-triangular representation unique?
Answer
  1. Write   as   and then   as  . If
     
    then,
     
    because  .
  2. We must first decide what the question means. Compare   with its restriction to the subspace  . The rangespace of the restriction is a subspace of  , so fix a basis   for this rangespace and extend it to a basis   for  . We want the relationship between these two.
     
    The answer falls right out of the prior item: if
     
    then the extension is represented in this way.
     
  3. Take   to be the span of  .
  4. Apply the answer from the second item to the third item.
  5. No. For instance  , projection onto the   axis, is represented by these two upper-triangular matrices
     
    where  .