Question

Please be clear.

2. Prove that the columns of a matrix A are linearly independent if and only if Ax = 0 has only the trivial solution. 3. Prove that any set of p vectors in R™ is linearly dependent if p > n.

Answer 1

are 2) Prove that columns of a matrix А linearly independent if and only if AX=0 has only the trivial solution of A Proof- first Assume that colums are linearly independent let diu A= diz 922 ain aan 929 be ami am2 amn mxn a mxn matzin. The columns of A are v = (al, ai, V2 = (012, 922, T ami ) -- amz) 2T 1 T. vn - (ain, azn, - amn) V1, V2 Vn) € IRM. Vn are linearly since V1, V2, independent . The equation has eiv+22V2 + txo no only trivial solution 1,= K2=- = Zn =0

But av,+R2V2+ tunin au E2, 1921 912 tnr a22 ain tan aan t- ܘ... ܘ ami Lamp amn dii 912 an 0 ni 22 ARI aar - den o Lami ama amn xn. = AX = 6 has only trivial solution. Conversely, suppose Ax=0 has only trivial solution. xiv + 2 V2 + tanno krivial solution. has only :: VI, V2, Un care linearly independent 3) Prove that the any set of vectors in 129 is linearly dependent if pen. proof: let hy, U2, • Up} be set veetors in Pn. of irn

consider xivi +22 V2 + top up = o This can be written fum in matin [ V2 Vn}x= 6 is a This & variables P system of equations. n variables . pan variables. are free ile tonow that has non .. excó solution. trivial where A= [vi V2 - Vn] hence at least one there. exists kisp. He to & hence the vectors Vn are linearly V, V2, dependent