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(Solved): Problem 6 Specify whether the following statement are TRUE or FALSE (no justification necessary). ( ...
Problem 6 Specify whether the following statement are TRUE or FALSE (no justification necessary). (a) Let
AinR^(n\times n). If
det(A)=0, a linear system
Ax=bnever has a solution. (b) Let
AinR^(n\times n). If the null space
N(A)={0}, then you can always find a unique solution for a linear system
Ax=b. (c)
AB=BAfor any square matrices
AinR^(n\times n)and
BinR^(n\times n). (d) For any vectors
x,yinR^(n)the following inequality holds:
||x||_(2)||y||_(2)<|x^(T)y|. (e) For any
AinR^(m\times n)and
BinR^(k\times n)the set
{xinR^(n):Ax=0,Bx=0}is a subspace. (f) Given nonzero vectors
xinR^(m)and
yinR^(n),dim(R(xy^(T)))=1. (g) For any
AinR^(m\times n)and
BinR^(m\times k)the set
{(yinR^(m):y=Ax):}for some
xinR^(n)or
y=Bzfor some
{:zinR^(k)}is a subspace. (h) For
AinR^(m\times n),||A||_(F)^(2)=tr(A^(T)A). BONUS If
dim(R(A))=mAx=bAinR^(m\times n),m, satisfies dim(R(A))=m, then you can always find a unique solution
for a linear system Ax=b.