Let v1 vector 1 2 3 v2 vector 1 1 4 v3 vector 3 3 2 and v

Let v_1 vector = (1, 2, 3), v_2 vector = (-1, 1, 4), v_3 vector = (3, 3, 2), and v_4 vector = (-2, -4, -6). Let W = span(v_1 vector, v_2 vector, v_3 vector, v_4 vector). Find a set of linearly independent vectors in R^3 which spans W (and, by doing this, find a basis therefore of W).

Solution

Let A =

1

-1

3

-2

2

1

3

-4

3

4

2

-6

We will reduce A to its RREEF as under:

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

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

Multiply the 2nd row by 1/3

Add -7 times the 2nd row to the 3rd row

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

Then the RREF of A is

1

0

2

-2

0

1

-1

0

0

0

0

0

Apparently, v₃ = 2v₁ –v₂ and v₄ = -2v₁ . Thus, v₁ and v₂ are the only linearly independent vectors in R³ which span W. Further a basis for W is { v₁,v₂}

1	-1	3	-2
2	1	3	-4
3	4	2	-6