Question

Linear Algebra: Let A be a 4x3 matrix and suppose that the vectors z1=(1,1,2)^T and (1,0,-1)^T form a basis for N(A). If the b = a1 + 2a2 + a3 (where b is a vector and a1, a2, a3 are column vectors of A), find all solutions of the system Ax = b.

Answer 1

There should be error.. if a=4*3 and dim(n(a)) =2 ,z1 shold be consist of 4 points...like z1=(a,s,0,t),z1=(a,s,f,0) if and only we can solve that problem is't it

Answer 2

if it has two free variable ..

Answer 3

where are you ...elliotc..

Answer 4

Am i correct....

Answer 5

please type something.....

Answer 6

this miight help:

let \(z_1=(1,1,2)^t\) and \(z_2=(1,0,-1)^t\) in \(\ker A=N(A)\). then
\[ \large 0=Az_1=a_1+a_2+2a_3 \]
and
\[ \large 0=Az_2=a_1-a_3 \].

Answer 7

Yaaa..something wrong with me...

Answer 8

from the second equation
\[ \large a_1=a_3 \]

Answer 9

so the first equation becomes
\[ \large 0=Az_1=3a_1+a_2 \]
so
\[ \large a_2=-3a_1 \]
therefore
\[ \large b=a_1+2a_2+a_3=a_1-6a_1+a_1=-4a_1 \]

Answer 10

so the system \(Ax=b\) actually is \(Ax=-4a_1\) so \(x=(-4,0,0)^t\)

Answer 11

i think that's it.