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underacheiver
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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