Let Bi = {ul, ua) and B2 = {vi, v2} be the bases for R2 in which ui = (2, 2), u2

Question

Let Bi = {ul, ua) and B2 = {vi, v2} be the bases for R2 in which ui = (2, 2), u2 = (4,-1), v-= (1.3), and v2 (-1,-1). (a) Find the transition matrix PB2Bi by row reduction. (b) Find the transition matrix PB, B2 by row reduction. (c) i Confirm that PB2B, another. and PB-^B, are inverses of one (d) Find the coordinate matrix for w = (5,-3) with respect to Bi, and use the matrix PBB to compute [wlB, from [WlB, (e) Find the coordinate matrix for w = (3,-5) with respect to B2, and use the matrix PB2-+Bi to compute [wlB, from [w]B

Explanation / Answer

34.(a).Let A =

1

-1

2

4

3

-1

2

-1

We can reduce A to its RREF as under:

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

Multiply the 2nd row by 1/2

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

Then the RREF of A is

1

0

0

-5/2

0

1

-2

-13/2

Thus, the transition matrix from B2 to B1 is PB2B1 =

0

-5/2

-2

-13/2

(b).Let M =

2

4

1

-1

2

-1

3

-1

We can reduce M to its RREF as under:

Multiply the 1st row by ½

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

Multiply the 2nd row by -1/5

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

Then the RREF of A is

1

0

13/10

-1/2

0

1

-2/5

0

Thus, the transition matrix from B1 to B2 is PB1B2 =

13/10

-1/2

-2/5

0

(c ). We have (PB2B1)( PB1B2)= I2 = (PB1B2) (PB2B1). Hence, PB2B1 and PB1B2 are inverses of each other.

(d). Let P =

2

4

5

2

-1

-3

The RREF of P is

1

0

-7/10

0

1

8/5

Therefore, [w]B1 = (-7/10,8/5)T. Then[w]B2 = (PB1B2) [w]B1 = (8,-2)T.

(e). Let Q =

1

-1

3

3

-1

-5

The RREF of Q is

1

0

-4

0

1

-7

Therefore, [w]B2 = (-4,-7)T. Then[w]B1 = (PB2B1) [w]B2 = (35/2,107/2)T.

1

-1

2

4

3

-1

2

-1

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