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## Homework Statement

I want to extend the below U set of vectors to R4.

u1 = (0, 0, 0, -4), u2 = (0, 0, -4, 3), u3 = (3, 2, 3, -2).

## The Attempt at a Solution

For a set of vectors to form a basis for Rn, the vectors must be LI and spans Rn(has n vectors)

u1, u2 and u3 are linearly independent and spans R3 but not R4.

To extend u1, u2 and u3 to R4. I require another vector, w. [itex]w\notin span{u1,u2,u3}[/itex]

One of the solution is trial-and-error which I am not keen.

I started with this:

0λ1 + 0λ2 +3λ3 +λ4w1= 0

0λ1 + 0λ2 +2λ3 +λ4w2= 0

0λ1 -4λ2 +3λ3 + λ4w3= 0

-4λ1 +3λ2 -2λ3 + λ4w4 = 0

which is a brick wall.

The other approach I had was UW = 0

I want to find the W set of vectors that maps the coefficient matrix U into the zero-vector.