The dimension of solution space of AT x 0 if a 4 times 6 ma

The dimension of solution space of A^T x = 0 if a 4 times 6 matrix A with real entries is a subspace of R^6. If {u, v, w} is a linearly independent set of vectors in a vector space V then the set of vectors {u, u + v, v + v+w} is also a linearly independent.

Solution

g)

A^T is a 6x4 matrix. Hence, x is of size 4x1

So the solution space is in R4 and not R6

h)

Let, a,b,c so that

au+b(u+v)+c(u+v+w)=0

(a+b+c)u+(b+c)v+cw=0

Sinc, u,v,w are linearly idnepdenet

a+b+c=0,b+c=0,c=0

This gives. a=b=c=0

Hence,

{u,u+v,u+v+w} is also a linearly independent set\'