(p) X{(x1, 22, 3)| x1- 2-23} is orthogonal to Y = {(y1,y2, y3) 1y1+y2+2y3T2. Let 1, 2, 3 be linearly independent vectors in R" and lety1 x1 ax2,Y2 = x2 + X3,and y3=3+x1Are y1, y2, 3y3 linearly independent? Prove your answer.3. Whether the following vectors x + 2,x +1,1 are linearly independent in P3.1 2 342 1-3 23 3 04. Let A =2Find a basis of N(A),/row space of A/ column space of A.(b) Find nullity (A) and rank A.5. Let A and B be two n xn matrices. Suppose that AB On. Show thatrank A + rank B

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Asked Nov 16, 2019
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(p) X
{(x1, 22, 3)| x1- 2-23} is orthogonal to Y = {(y1,y2, y3) 1y1+y2+2y3
T
2. Let 1, 2, 3 be linearly independent vectors in R" and let
y1 x1 ax2,
Y2 = x2 + X3,
and y3=3+x1
Are y1, y2, 3y3 linearly independent? Prove your answer.
3. Whether the following vectors x + 2,x +1,
1 are linearly independent in P3.
1 2 3
4
2 1-3 2
3 3 0
4. Let A =
2
Find a basis of N(A),/row space of A/ column space of A.
(b) Find nullity (A) and rank A.
5. Let A and B be two n xn matrices. Suppose that AB On. Show that
rank A + rank B <n.
6. Let E = [x2, x, 1] and F = x2 +2x, 1, 1] be ordered bases of P3.
(a) Find the transition matrix representing the change from E to F
x2 -2
(b) Use the transition matrix to find coordinate vector [plF with p
1
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(p) X {(x1, 22, 3)| x1- 2-23} is orthogonal to Y = {(y1,y2, y3) 1y1+y2+2y3 T 2. Let 1, 2, 3 be linearly independent vectors in R" and let y1 x1 ax2, Y2 = x2 + X3, and y3=3+x1 Are y1, y2, 3y3 linearly independent? Prove your answer. 3. Whether the following vectors x + 2,x +1, 1 are linearly independent in P3. 1 2 3 4 2 1-3 2 3 3 0 4. Let A = 2 Find a basis of N(A),/row space of A/ column space of A. (b) Find nullity (A) and rank A. 5. Let A and B be two n xn matrices. Suppose that AB On. Show that rank A + rank B <n. 6. Let E = [x2, x, 1] and F = x2 +2x, 1, 1] be ordered bases of P3. (a) Find the transition matrix representing the change from E to F x2 -2 (b) Use the transition matrix to find coordinate vector [plF with p 1

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Expert Answer

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Step 1

Consider the given matrix A as,

1
2 3
4
A =2
1 -3-2
3 3
0
2
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1 2 3 4 A =2 1 -3-2 3 3 0 2

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Step 2

On solving matrix A to reduce in row-echelon form,

1
3 4
1
2
3
4
2
0 -3
A =2
-9 -10
1
-3 -2
0 -3
-9 -10
3
0
2
Г1 2 3
1
R3-R-R
2
4
R2)R
4
о -3 -9 -10
0 3
9 10
0 0
0 0 0 0
0
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1 3 4 1 2 3 4 2 0 -3 A =2 -9 -10 1 -3 -2 0 -3 -9 -10 3 0 2 Г1 2 3 1 R3-R-R 2 4 R2)R 4 о -3 -9 -10 0 3 9 10 0 0 0 0 0 0 0

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Step 3

General solution of...

1 2 3
4
1 2 3
4
10
A 0 3 9 10 0 1 3
0 0 0
0 0 0
0
Then AX 0 implies, x 2x, +3x3 +4x4 = 0 and 3x2 +9x3 +10x = 0
Let, x kand x = t
9k10
9k+10t
2
(9k +10t)
Then x
and x 3k-4t 2
3
-3k -4t
9k-12t+18k+20t 9k+8t
3
Then solution in vector form is,
9k+8t
9k+8t
8
3
9k+ 10
-(9k 10t)
-10
-9
= k
+t
3
3k
3
0
k
3t
0
t
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1 2 3 4 1 2 3 4 10 A 0 3 9 10 0 1 3 0 0 0 0 0 0 0 Then AX 0 implies, x 2x, +3x3 +4x4 = 0 and 3x2 +9x3 +10x = 0 Let, x kand x = t 9k10 9k+10t 2 (9k +10t) Then x and x 3k-4t 2 3 -3k -4t 9k-12t+18k+20t 9k+8t 3 Then solution in vector form is, 9k+8t 9k+8t 8 3 9k+ 10 -(9k 10t) -10 -9 = k +t 3 3k 3 0 k 3t 0 t

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