Given vector v1 2 3 5 4 v2 3 3 1 1 and v3 0 5 1 4 sum R4

Given vector v_1 = (-2 -3 5 -4), v_2 = (-3 3 1 -1), and v_3 = (0 5 -1 -4) sum R^4, v = (-5 0 18 -15) sum span (v_1, v_2, v_3), find coefficients a_1, a_2, a_3 such that a_1 v_1 + a_2 v_2 + a_3 v_3 = v.

Solution

Let A =

-2

-3

0

-15

-3

3

5

0

5

1

-1

18

-4

-1

-4

-15

We will reduce A to its RREF as under:

Multiply the 1st row by -1/2

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

Add -5 times the 1st row to the 3rd row

Add 4 times the 1st row to the 4th row

Multiply the 2nd row by 2/15     

Add 13/2 times the 2nd row to the 3rd row

Add -5 times the 2nd row to the 4th row

Multiply the 3rd row by 3/10

Add 22/3 times the 3rd row to the 4th row

Add -2/3 times the 3rd row to the 2nd row

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

Then the RREF of A is

1

0

0

3

0

1

0

3

0

0

1

0

0

0

0

0

Thus, v = 3v₁+3v₂ +0v₃; a₁ = 3,a₂ = 3 and a₃ = 0.

-2	-3	0	-15
-3	3	5	0
5	1	-1	18
-4	-1	-4	-15