Prove that Col(A) is a proper subset of Nul(A)

[junsugal]

Homework Statement

Prove that if A is a nxn matrix such that A^2=0, then Col(A) is a proper subset of Nul(A)

Homework Equations

The Attempt at a Solution

None, i have no idea how to start.
Please guide me or explain what does the question means and how to approach.

 

[tiny-tim]

hi junsugal!

(do you mean Ker(A)? it's spelled "kernel", not "colonel" … a kernel is the soft part inside the shell of a nut! )

it means you have to prove that there exists an x such that x is in Nul(A) but not Col(A)

(i think the question should have stipulated that A is non-zero)

 

[junsugal]

hi junsugal!

(do you mean Ker(A)? it's spelled "kernel", not "colonel" … a kernel is the soft part inside the shell of a nut! )

it means you have to prove that there exists an x such that x is in Nul(A) but not Col(A)

(i think the question should have stipulated that A is non-zero)

Hi :)

It is Col(A).
I thought it means that Nul(A) is a subspace of Col(A)?
well, I'm still confused though.
I tried to solve this problem asuming that A is zero matrix.
Because I couldn't think of any nxn matrix that will get zero after multiply by itself.

 

[tiny-tim]

I thought it means that Nul(A) is a subspace of Col(A)?

no …

… Col(A) is a proper subset of Nul(A)

… means that Col is a subset of Nul, but is less than Nul

I tried to solve this problem asuming that A is zero matrix.

no!

A must be non-zero