Find the transition matrix corresponding to the change of ba

Find the transition matrix corresponding to the change of basis from

Let v_1 = (3, 1, 1)^T v_2 = (0, 1, 0)^T, v_3 = (0, 1, 1)^T, and let u_1 = (1, 2, 1)^T, u_2 = (2, 5, 2)^T, u_3 = (4, 8, 5)^T. (Note that [u_1, u_2, u_3] is the basis from the previous exercise.) (a) Find the transition matrix corresponding to the change of basis from [v_1, v_2, v_3] to [u_1, u_2, u_3]. (b) If x = 2v_1 + 3v_2 - 4v_3, express x as a linear combination of u_1, u_2, u_3: x = u_1+ u_2+ u_3.

Solution

1

2

4

3

0

0

2

5

8

1

1

1

1

2

5

1

0

1

We will reduce A to its RREF as under:

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

Add -1 times the 1st row to the 3rd row

Add -4 times the 3rd row to the 1st row

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

Then the RREF of A is

1

0

1

21

-2

-6

0

1

0

-5

1

1

0

0

1

-2

0

1

Then the required change of basis matrix is

21

-2

-6

-5

1

1

-2

0

1

Also, v₁ = 21u₁-5u₂-2u₃, v₂ = -2u₁+u₂ and v₃ = -6u₁ +u₂ +u₃

(b) If x = 2v₁+3v₂-4v₃, then x = 2(21u₁-5u₂-2u₃ )+3(-2u₁+u₂ ) -4(-6u₁ +u₂ +u₃ ) = 60u₁ -11u₂ -5u₃

1	2	4	3	0	0
2	5	8	1	1	1
1	2	5	1	0	1