Linear Algebra/Sums and Scalar Products/Solutions

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Solutions[edit]

This exercise is recommended for all readers.
Problem 1

Perform the indicated operations, if defined.

Answer
  1. 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. Matrices have an additive inverse .
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.

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.