Question

Recall that if T: R" R" is a linear transforrmation T(x) = [Tx, where [T is the transformation matrix, then 1. ker(T) null([T] (ker(T) is the kernel of T) 2. T is one-to-one exactly when ker(T) = {0 3. range of T subspace spanned by the columns of [T] col([T) 4. T is onto exactly when T(x) = [Tx = b is consistent for all b in R". 5. Also, T is onto exactly when range of T col([T]) = R" (codomain of T) R4 bethe matrix transformation Let T: R4 -1 1 0 2 X 2 -3 1-5 T 2-2 1-3 Z 1 -1 3 h u Answer the following questions True or False. Make sure to justify your answer. Choose the most appropriate answer!!! Choose. T is onto when h 2 Choose... T is 1-1 when h 5 True False since rank([T]) < # of columns of [T] The kernel of T consists only of the zero vector when h = 3True since rank([T]) < # of columns of [T] True since rank([T]) = # of columns of [T) True since rank([T]) = # of rows of [T] False since rank([T]) < # of rows of [T] False Choose T is 1-1 when h 1 range of T equals R4 when h 1. Choose The kernel of T contains non-zero vectors when h = 1.

Answer 1

Let A =

-1	1	0	2
2	-3	1	-5
2	-2	1	-3
1	-1	3	h

The RREF of A is

1	0	0	0
0	1	0	0
0	0	1	0
0	0	0	h-1

1. TRUE. When h = 2, the RREF of A is I₄ so that the columns of A are linearly independent and span R⁴. Hence T is onto.

2. TRUE. When h = 5, the RREF of A is I₄ so that the columns of A are linearly independent. Hence ker(T) = {0} so that T is one-to-one.

3. TRUE. When h = 3, the RREF of A is I₄ so that the columns of A are linearly independent. Hence ker(T) = {0}.

4. FALSE. When h = 1, the RREF of A is

1	0	0	0
0	1	0	2
0	0	1	1
0	0	0	0

Here, the columns of A are not linearly independent. Hence ker(T) ≠ {0} so that T is not one-to-one.

5. TRUE. When h ≠ 1, the columns of A are linearly independent and span R⁴. Hence the range of T equals R⁴.

6. TRUE. Please see part 4 above.