TRUE or FALSE If true show if false give a counter example o

TRUE or FALSE. If true show; if false, give a counter example or explain.

f) The dimension of solution space of Ax=0 is 1, where A=

1 2 3

0 1 4

0 0 1

g) The dimension of solution space of (A^T)x=0 if a 4x6 matrix A with real entries is a subspace of R6.

h) If {u, v, w} is a linearly independent set of vectors in a vector space V then the set of vectors {u,u+v,u+v+w} is also linearly independent.

False.

Clearly the rows of A are linearly indepdenet. HEnce rank of A=3

For A rank nullity theorem gives

rank(A)+nullity(A)=3

rank(A)=3 ,hence nullity(A)=0

ie dim ker(A)=0

Hence, ker(A)={0}

ie dimensino of solution space of Ax=0 =0

A^T is of size 6x4 because A is of size:4x6

Hence,x is of size:4x1 and columns of A are of size: 6x1 ie solution space is a subpsace of R6

Let, a,b,c so that

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

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

u,v,w are linearly independent

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

giving, a=b=c=0

Hence,

{u,u+v,u+v+w} is also linearly indepdent