LInear Algebra Help Please Show all work thank you Consider

LInear Algebra Help, Please. Show all work, thank you.

Consider the matrix A = [1 3 4 2 1 3 4 2] Find a basis for Col A. Find a basis for Nul A. Find two elements in the null Nul A.

Solution

1)

BOth the rows are identical so rank(A)=1=dim(col(A))=1

Basis col(A)=first column of A

2)

A is 2x4 matrix

So, rank(A)+nullity(A)=4

HEnce nullity=3

ie dim(null(A))=3

Ax=0

let, x=[r s t u]

So

Ax=0 gives

r+3s+4t+2u=0

r=-3s-4t-2u

x=[-3s-4t-2u,s,t,u]

x=s[-3,1,0,0]+t[-4,0,1,0]+u[-2,0,0,1]

Hence basis for Nul A is

{[-3,1,0,0],[-4,0,1,0],[-2,0,0,1]}

3.

Two elements are

[-3,1,0,0] and [-2,0,0,1]