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6.E: Exercises for Chapter 6

  2. Last updated
    Mar 5, 2021
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( \newcommand{\kernel}{\mathrm{null}\,}\)

Calculational Exercises

1. Define the map by .

  1. Show that is linear.
  2. Show that is surjective.
  3. Find .
  4. Find the matrix for with respect to the canonical basis of .
  5. Find the matrix for with respect to the canonical basis for the domain and the basis for the target space .
  6. Show that the map given by is not linear.

2. Let be defined by

  1. Show that is surjective.
  2. Find .
  3. Find the matrix for with respect to the canonical basis of .
  4. Show that the map given by is not linear.

3. Consider the complex vector spaces and with their canonical bases, and define be the linear map defined by , where is the matrix

Find a basis for

4. Give an example of a function having

the property that

but such that is not a linear map.

5. Show that the linear map is surjective if

6. Show that no linear map can

have as its null space the set

7. Describe the set of solutions of the system of equations

Proof-Writing Exercises

1. Let and be vector spaces over with finite-dimensional, and let be any
subspace of . Given a linear map prove that there exists a linear map
such that, for every

2. Let and be vector spaces over and suppose that is injective.
Given a linearly independent list of vectors in , prove that the
list is linearly independent in

3. Let and be vector spaces over and suppose that the linear maps
and are both injective. Prove that the composition map is injective.

4. Let and be vector spaces over and suppose that is surjective.
Given a spanning list for , prove that



5. Let and be vector spaces over with finite-dimensional. Given
prove that there is a subspace of such that

6. Let be a vector space over and suppose that there is a linear map
such that both and are finite-dimensional subspaces of . Prove that
must also be finite-dimensional.

7. Let and be finite-dimensional vector spaces over with and
Prove that

8. Let be a finite-dimensional vector space over with Prove that
is invertible if and only if both and are invertible.

9. Let be a finite-dimensional vector space over with and denote by
I the identity map on . Prove that if and only if

This page titled 6.E: Exercises for Chapter 6 is shared under a not declared license and was authored, remixed, and/or curated by Isaiah Lankham, Bruno Nachtergaele, & Anne Schilling.

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