Question

Question 5 For each given vector b and matrix A, determine if b e im(A) 1 -2 3 (a) b 0 A 21 3 0 5 15 (b) b A2-24 9 Question 6 Find the rank and nullity of the given linear transformations T and determine which are one-to-one and which are onto. r+ y ri+r2 Question 7 Find nullity(T) if (a) T:R R2, rank(T) 1 (b) T:RR, rank(T) 0 (c) T : Rs ? R2, rank(T)-1 Question 8 Let T : V ? V be a linear transformation and assume din(V) n. If ker(T) = 0) which of the following statements are true and which are false. (a) rank(T)-n-1 (b) inn(T) = V (c) nullity(T)=0 (d) T is onto (e) T(u) = T(v) if and only if u v for all u, v E V

Answer 1

5. (a). Let M = [A|b] =

1	-2	3	11
2	-1	3	10
4	1	-1	4

The RREF of M is

1	0	0	2
0	1	0	-3
0	0	1	1

It implies that b = (11,10,4)^T = 2(1,2,4)^T-3(-2,-1,1)^T+(3,3,-1)^T. Hence b ? im(A) = col(A).

(b). Let M = [A|b] =

0	5	15	9
2	-2	-4	-2
3	-3	-6	-4

The RREF of M is

1	0	1	0
0	1	3	0
0	0	0	1

It implies that b = (9,-2,-4)^T ? im(A) = col(A).

6. (a). T(e₁) = T(1,0)^T = (2,0)^T and T(e₂) = T(0,1)^T = (0,3)^T. Hence the standard matrix of T is A =

2	0
0	3

The RREF of A is I₂ which implies that the columns of A are linearly independent. Hence T is one-to-one.

(b). T(e₁) = T(1,0,0)^T =(1,0)^T,T(e₂) = T(0,1,0)^T =(1,0)^T and T(e₃) = T(0,0,1)^T =(0,1)^T. Hence the standard matrix of T is A =

1	1	0
0	0	1

The columns of A are not linearly independent. Hence T is not one-to-one.

(c ). T(e₁) = T(1,0)^T =(1,2,0)^T, T(e₂) = T(0,1)^T =(1,1,1)^T. Hence the standard matrix of T is A =

1	1
2	1
0	1

The RREF of A is

1	0
0	1
0	0

It implies that the columns of A are linearly independent. Hence T is one-to-one.

(d). T(e₁) = T(1,0,0,0)^T =(1,0,0)^T, T(e₂) = T(0,1,0,0)^T =(1,1,0)^T T(e₃) = T(0,0,1,0)^T =(0,1,1)^T T(e₄) = T(0,0,0,1)^T =(0,0,1)^T. Hence the standard matrix of T is A =

1	1	0	0
0	1	1	0
0	0	1	1

A has 4 column vectors. Since dim(R³)= 3, hence the columns of are not linearly independent. Hence T is not one-to-one.

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