Suppose A is a 3 times 3 matrix and the vectors u 1 2 1 v

Suppose A is a 3 times 3 matrix and the vectors u = [1 2 1], v = [2 -1 0] span the subspace defined by A_x = 0. Determine if the vector w = [-1 1 1] is in Span(u, v). Will A w = 0? Solution: Det [1 2 -1 2 -1 1 1 0 1] = -1 - 2(2 - 1) - 1(-1) = -1 -2 + 1 = 2 notequalto 0 So the columns are linearly independent, which means that w cannot be made equal to any linear combination of the other vectors.

Solution

Let, w=au+bv

So, 1=a

-1=a+2b

1=2a-b

Hence no solution

So, w is not in span{u,v}

No. Aw is not equal to 0

Because otherwise w would be in the subspace Ax=0 which is spanned by u and v