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## Homework Statement

1. Which of the following is not a linear transformation from 3 to 3?

a. T(x, y, z) = (x, 2y, 3x - y)

b. T(x, y, z) = (x - y, 0, y - z)

c. T(x, y, z) = (0, 0, 0)

d. T(x, y, z) = (1, x, z)

e. T(x, y, z) = (2x, 2y, 5z)

2. Which of the following statements is not true?

a. If A is any n × m matrix, then the transformation T: defined by T(x) = Ax is always a linear transformation.

b. If T: U → V is any linear transformation from U to V then T(xy) = T(x)T(y) for all vectors x and y in U.

c. If T: U → V is any linear transformation from U to V then T(-x) = -T(x) for all vectors x in U.

d. If T: U → V is any linear transformation from U to V then T(0) = 0 in V for 0 in U.

e. If T: U → V is any linear transformation from U to V then T(2x) = 2T(x) for all vectors x in U.

3. If T: U → V is any linear transformation from U to V then

a. the kernel of T is a subspace of U

b. the kernel of T is a subspace of V

c. the range of T is a subspace of U

d. V is always the range of T

e. V is the range of T if, and only if, ket T = {0}

4. If T: U → V is any linear transformation from U to V and B = {u 1, u 2, ..., u n} is a basis for U, then set T(B) = {T(u 1), T(u 2), ... T(u n)}

a. spans V

b. spans U

c. is a basis for V

d. is linearly independent

e. spans the range of T

5. P 3 is a vector space of polynomials in x of degree three or less and Dx(p(x)) = the derivative of p(x) is a transformation from P 3 to P 2.

a. the nullity of Dx is two.

b. The polynomial 2x + 1 is in the kernel of Dx.

c. The polynomial 2x + 1 is in the range of Dx.

d. The kernel of Dx is all those polynomials in P 3 with zero constant term.

e. The rank of Dx is three.

6.Let Ax = b be the matrix representation of a system of equations. The system has a solution if, and only if, b is in the row space of the matrix A.

a. True

b. False

7.If A is an n × n matrix, then the rank of A equals the number of linearly independent row vectors in A.

a. True

b. False

## Homework Equations

## The Attempt at a Solution

1. d

2. b

3. a

4. a

5. d

6. b

7. a