Linear Algebra/Sums and Scalar Products/Solutions

SolutionsEdit

This exercise is recommended for all readers.
Problem 1

Perform the indicated operations, if defined.

  1.  
  2.  
  3.  
  4.  
  5.  
Answer
  1.  
  2.  
  3.  
  4.  
  5. Not defined.
Problem 2

Prove Theorem 1.5.

  1. Prove that matrix addition represents addition of linear maps.
  2. Prove that matrix scalar multiplication represents scalar multiplication of linear maps.
Answer

Represent the domain vector   and the maps   with respect to bases   in the usual way.

  1. The representation of  
     
     
    regroups
     
    to the entry-by-entry sum of the representation of   and the representation of  .
  2. The representation of  
     
     
    is the entry-by-entry multiple of   and the representation of  .
This exercise is recommended for all readers.
Problem 3

Prove each, where the operations are defined, where  ,  , and   are matrices, where   is the zero matrix, and where   and   are scalars.

  1. Matrix addition is commutative  .
  2. Matrix addition is associative  .
  3. The zero matrix is an additive identity  .
  4.  
  5.  
  6. Matrices have an additive inverse  .
  7.  
  8.  
Answer

First, each of these properties is easy to check in an entry-by-entry way. For example, writing

 

then, by definition we have

 

and the two are equal since their entries are equal  . That is, each of these is easy to check by using Definition 1.3 alone.

However, each property is also easy to understand in terms of the represented maps, by applying Theorem 1.5 as well as the definition.

  1. The two maps   and   are equal because  , as addition is commutative in any vector space. Because the maps are the same, they must have the same representative.
  2. As with the prior answer, except that here we apply that vector space addition is associative.
  3. As before, except that here we note that  .
  4. Apply that  .
  5. Apply that  .
  6. Apply the prior two items with   and  .
  7. Apply that  .
  8. Apply that  .
Problem 4

Fix domain and codomain spaces. In general, one matrix can represent many different maps with respect to different bases. However, prove that a zero matrix represents only a zero map. Are there other such matrices?

Answer

For any   with bases  , the (appropriately-sized) zero matrix represents this map.

 

This is the zero map.

There are no other matrices that represent only one map. For, suppose that   is not the zero matrix. Then it has a nonzero entry; assume that  . With respect to bases  , it represents   sending

 

and with respcet to   it also represents   sending

 

(the notation   means to double all of the members of D). These maps are easily seen to be unequal.

This exercise is recommended for all readers.
Problem 5

Let   and   be vector spaces of dimensions   and  . Show that the space   of linear maps from   to   is isomorphic to  .

Answer

Fix bases   and   for   and  , and consider   associating each linear map with the matrix representing that map  . From the prior section we know that (under fixed bases) the matrices correspond to linear maps, so the representation map is one-to-one and onto. That it preserves linear operations is Theorem 1.5.

This exercise is recommended for all readers.
Problem 6

Show that it follows from the prior questions that for any six transformations   there are scalars   such that   is the zero map. (Hint: this is a bit of a misleading question.)

Answer

Fix bases and represent the transformations with   matrices. The space of matrices   has dimension four, and hence the above six-element set is linearly dependent. By the prior exercise that extends to a dependence of maps. (The misleading part is only that there are six transformations, not five, so that we have more than we need to give the existence of the dependence.)

Problem 7

The trace of a square matrix is the sum of the entries on the main diagonal (the   entry plus the   entry, etc.; we will see the significance of the trace in Chapter Five). Show that  . Is there a similar result for scalar multiplication?

Answer

That the trace of a sum is the sum of the traces holds because both   and   are the sum of   with  , etc. For scalar multiplication we have  ; the proof is easy. Thus the trace map is a homomorphism from   to  .

Problem 8

Recall that the transpose of a matrix   is another matrix, whose   entry is the   entry of  . Verifiy these identities.

  1.  
  2.  
Answer
  1. The   entry of   is  . That is also the   entry of  .
  2. The   entry of   is  , which is also the   entry of  .
This exercise is recommended for all readers.
Problem 9

A square matrix is symmetric if each   entry equals the   entry, that is, if the matrix equals its transpose.

  1. Prove that for any  , the matrix   is symmetric. Does every symmetric matrix have this form?
  2. Prove that the set of   symmetric matrices is a subspace of  .
Answer
  1. For  , the   entry is   and the   entry of is  . The two are equal and thus   is symmetric. Every symmetric matrix does have that form, since it can be written  .
  2. The set of symmetric matrices is nonempty as it contains the zero matrix. Clearly a scalar multiple of a symmetric matrix is symmetric. A sum   of two symmetric matrices is symmetric because   (since   and  ). Thus the subset is nonempty and closed under the inherited operations, and so it is a subspace.
This exercise is recommended for all readers.
Problem 10
  1. How does matrix rank interact with scalar multiplication— can a scalar product of a rank   matrix have rank less than  ? Greater?
  2. How does matrix rank interact with matrix addition— can a sum of rank   matrices have rank less than  ? Greater?
Answer
  1. Scalar multiplication leaves the rank of a matrix unchanged except that multiplication by zero leaves the matrix with rank zero. (This follows from the first theorem of the book, that multiplying a row by a nonzero scalar doesn't change the solution set of the associated linear system.)
  2. A sum of rank   matrices can have rank less than  . For instance, for any matrix  , the sum   has rank zero. A sum of rank   matrices can have rank greater than  . Here are rank one matrices that sum to a rank two matrix.