1 pt Consider the ordered bases B 3 4x 3x 2 and C 2 4x for t

(1 pt Consider the ordered bases B (3 4x, 3x 2) and C (2, 4x) for the vector space P2Lx

Solution

e. Let A =

3

-2

3

-4

3

-2

In order to determine the coordinates of p(x) = 3-2x in the ordered basis B, we will reduce A to its RREF as under:

Multiply the 1st row by 1/3

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

Multiply the 2nd row by 3

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

Then the RREF of A is

1

0

5

0

1

6

Therefore, p(x) = 3-2x = 5(3-4x)+6(3x-2) and [p(x)]_B= (5,6)^T.

f. The coordinate vector of q(x) in the ordered basis C is (1,1)^T. Therefore, q(x) = 2+4x. Let M =

3

-2

2

-4

3

4

In order to determine the coordinates of p(x) = 3-2x in the ordered basis B, we will reduce M to its RREF as under:

multiply the 1st row by 1/3         

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

Multiply the 2nd row by 3

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

Then the RREF of M is

1

0

14

0

1

20

Therefore, q(x) = 2+4x = 14(3-4x)+20(3x-2) and [q(x)]_B= (14,20)^T.

3	-2	3
-4	3	-2