## Instructions

There are 15 problems on this exam, and you only need to turn in 6 of them. You may choose any 6, and you may consult your notes, our textbook, and me while working on them. You are not allowed to use any other resources or discuss your work with any other people. The exam is due on Tuesday, April 5th.

Note: each question is worth 20 points.

Remember to write clearly and to show your work. Enjoy!

## The Actual Exam

**Question 1**. Prove that if is an isomorphism, then .

**Question 2**. Let be a basis for , and let denote the standard basis for this space.

(a) Explain why the matrix representation of the identity map, , has as its matrix representation (relative to the bases for the domain and for the codomain)

(b) Let and suppose is a basis for . Moreover, also suppose that each vector in this basis is an ** eigenvector **for the homomorphism , i.e. suppose that

where the scalars . Explain / prove that is the diagonal matrix

**Question 3**. Let be the function

.

(a) Prove that is linear.

(b) Prove that is one-to-one, or find two inputs in that have the same output.

(c ) Prove that is onto, or find an element in the codomain that is not in the range space of .

(d) Compute the matrix representation of relative to the bases for the domain and for the co-domain.

**Question 4**. Let be an matrix.

(a) The **column space** of is the subspace of that is spanned by the columns of (where each column is viewed as a column vector in ).

Prove that the column space of equals the *range space* of . Note that the range space of a matrix is the set of outputs

where multiplies against vectors via matrix multiplication.

(b) The **row space** of is the subspace of that is spanned by the rows of (where each row is viewed as a vector in ). Explain / prove that the dimension of the row space always equals the dimension of the column space.

**Question 5**. Suppose is an isomorphism. Prove that if is a basis for , then is a basis for .

**Question 6**. Consider the 4-dimensional vector space and the function given by , i.e.

.

(a) Prove that is linear.

(b) Is one-to-one? Prove your answer.

(c ) Compute the rank of .

(d) Compute the matrix representation of relative to the standard basis for (for both the domain and co-domain).

**Question 7**. Let and be finite-dimensional vector spaces with bases and , respectively. Prove that the function given by

is an isomorphism of vector spaces.

**Question 8**. Let be the function where is the matrix

.

(a) Find a basis for the range space, .

(b) Find a basis for the null space, .

(c ) Compute the determinant of .

(d) Are the columns of linearly independent vectors? Justify your answer.

(e) Are the rows of linearly independent vectors? Justify your answer.

**Question 9**. Let be an matrix.

(a) Prove that is non-singular .

(b) Prove that is non-singular

**Question 10**. Consider the function given by

.

(a) Just to make sure you understand the function , compute the value of

.

(b) Prove that .

(c ) Find a basis for the null space of .

(d) Use only your work in part (c ) to compute the rank of .

(e) Is the polynomial in the range space ? Explain your answer.

**Question 11**. Let be a square matrix.

(a) What does it mean to say that a vector is an eigenvector (with eigenvalue ) for ?

(b) Let be an eigenvalue for the matrix , and define the set

.

Prove that is a subspace of .

(c ) Prove that .

**Question 12**. (a) Suppose is a homomorphism and that

.

Find a formula for .

(b) Instead, suppose is a homomorphism and that

where is the basis

Find a formula for .

**Question 13**. (a) Is the set of vectors

a basis for ? Prove your answer.

(b) Is the set of vectors

a basis for ? Prove your answer.

(c ) Is the matrix

one-to-one? Prove your answer.

(d) Is the matrix

onto? Prove your answer.

(e) Is the set of vectors

a basis for ? (Here is the same matrix as in parts (c ) and (d).) Prove your answer.

(f) Is the determinant of zero? Prove / explain your answer.

**Question 14**. Let be the matrix

and consider the set of all matrices that “commute” with ; that is, consider the set of matrices

.

(a) Prove that is a subspace of .

(b) Compute the dimension of .

(c ) Find a matrix so that or explain why no such matrix exists.

**Question 15**. Assume that

(a) Prove that the range space is a subspace of the codomain .

(b) Prove that the null space is a subspace of .

(c ) Prove that if is a subspace of the domain , then the set (called “the image of under “) is a subspace of the codomain . This generalizes the notion of range space.

(d) Suppose is a subspace of the codomain . Is it true that the set is a subspace of ? Prove your answer.