Decide N(F) and V(F) for Linear mapping in basis w. Matrix


The matrix is
(1 2 ).
(3 6 )
Call it A

A 2 x 2 matrix its located in some basis
E = (e1,e2)

I want to find N(F) and V(F)

1. So I will start by findin N(F)

What does that mean?

Well
F(u) = zero vector

I want to know which vectors which will
Fall on the zero vector

This means it's the same as

AX=0

It would lead to a linear equation system

X1 + 2x2 = 0
3x1 + 6x2 = 0


So I suppose I should do gauss

If I eliminate 3x1 from row 2
I will get all of the 2nd row to become

0=0 if I multiplied row 1 with -3

Getting

0=0 leaves me wondering what to do next?


2. I guess V(F) is gotten by cross multiplying
Two column vectors of the matrix A
This also leaves me wondering because it's a
2x2 matrix

Could someone show me how to solve it?