solve the problem a 3b 5d c d Let H a b c d in R 3a 9b 4c 11

solve the problem a 3b 5d c d Let H a, b, c, d in R -3a 9b 4c 11d Find the dimension of the subspace H. A dim H 1 B dim H 4 C dim H 3 D dim H 2

Solution

1. We have H = span{ v} where v = (a+3b+5d, c+d, --3a-9b+4c,-11d, -c-d)^T . Let us perform the following row operations on v:

Add 3times row 1 to row3; Add 1 time row 4 to row 2

Multiply row 3 by ¼; Add 1 time row 3 to row 4.

Then v changes to u = ( a+3b+5d, 0, c+d, 0)^T = a(1,0,0,0)^T+3b(1,0,0,0)^T+c(0,0,1,0)^T+ d(5,0,1,0)^T so that H = span{(1,0,0,0)^T,(0,0,1,0)^T,(5,0,1,0)^T }. Thus, the dimension of H is 3. Option C is the correct answer.

2. Let A =

1

-4

1

2

3

-10

5

4

Let us reduce A to its RREF as under:

Add -3 times the 1st row to the 2nd row

Multiply the 2nd row by ½

Add 4 times the 2nd row to the 1st row

Then the RREF of A is

1

0

5

-2

0

1

1

-1

The change of coordinates matrix from B to C is

5

-2

1

-1

Option D is the correct answer.

3. Since b₁ = c₁-5c₂ and b₂ = 3c₁-2c₂, the change of basis matrix from B to C is

1

-5

3

-2

Option D is the correct answer:

1	-4	1	2
3	-10	5	4