Linear algebra - basis multiple choice questions

[underacheiver]

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

 

[Billy Bob]

4 and 5 are wrong

 

[underacheiver]

How do you figure out the right answer??
4. if it doesn;t span U then does it span V?
5. is the answer b? because i don't see how it can be anything else then.

 

[Billy Bob]

Play with examples. For #4, use U=the plane, V=the plane, and B={i,j} (the standard basis). Can you give an example of T that isn't one-to-one? Now test each of the five responses with this T. I bet you can rule out four of them.

For #5, write down an actual cubic, and then find Dx of your cubic. Try another one. Pretty soon, you'll find that one of the five responses is obviously correct (and hopefully the other four are therefore wrong).

 

[underacheiver]

i finally got b for 4 and e for 5
for 4, b completes the theorum in one of my textbooks.
for 5, i solved the nullity of Dx as 0, thus the rank has to be 3? is that correct?