Hello Suppose that v1 2103 v2 3 152 and v3 1021 Which of

Hello,

Suppose that v1 = (2,1,0,3), v2 = (3, 1,5,2), and v3 = (1,0,2,1). Which of the following vectors are in span {v1, v2, v3 }?

(2, 3, 7, 3)

(0,0,0,0)

(1,1,1,1)

(4, 6, 13, 4)

Please use matrices (show working) to solve.

Solution

To determine whether w belongs to span (v1; v2; v3), we are to look to write w as a linear
combination of v1, v2, v3. For this purpose, we need to ¯nd three scalars c1; c2; c3, such that
w = c1v1 + c2v2 + c3v3.

This amounts to solve the system Ac = w for c = (c1; c2; c3)T , where
matrix A = (v1 v2 v3).

Now apply Gaussian to reduce the augmented matrix in the echelon form:

Find the pivot in the 1st column (inversing the sign in the whole row) and swap the 3rd and the 1st rows

Multiply the 1st row by 3

Subtract the 1st row from the 2nd row and restore it

Multiply the 1st row by 2

Subtract the 1st row from the 3rd row and restore it

Find the pivot in the 2nd column in the 2nd row (inversing the sign in the whole row)

Subtract the 2nd row from the 3rd row

Make the pivot in the 3rd column by dividing the 3rd row by 15

Multiply the 3rd row by -2

Subtract the 3rd row from the 1st row and restore it

Multiply the 3rd row by -11

Subtract the 3rd row from the 2nd row and restore it