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

• junsugal
In summary: Col(A) = Nul(A) = {0})In summary, to prove that if A is a nxn matrix such that A^2=0, then Col(A) is a proper subset of Nul(A), you need to show that there exists an x in Nul(A) that is not in Col(A). This means that Col(A) is a subset of Nul(A), but is less than Nul(A). Additionally, A must be non-zero, as if it is the zero matrix, then Col(A) and Nul(A) are equal.
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)

## 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.

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)

tiny-tim said:
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.

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

no …
junsugal said:
… 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

## What does it mean for Col(A) to be a proper subset of Nul(A)?

Being a proper subset means that Col(A) is a subset of Nul(A), but not equal to it.

## What is the relationship between Col(A) and Nul(A)?

Col(A) and Nul(A) are both subspaces of the vector space of A. They are complementary subspaces, meaning that their intersection is only the zero vector.

## How can you prove that Col(A) is a proper subset of Nul(A)?

To prove this, we need to show that every vector in Col(A) is also in Nul(A), but there exists at least one vector in Nul(A) that is not in Col(A).

## What is the significance of Col(A) being a proper subset of Nul(A)?

This relationship tells us that the column space and null space of a matrix are not equal, and that there exists a non-trivial solution to the equation Ax=0.

## Can you give an example of a matrix where Col(A) is a proper subset of Nul(A)?

Yes, consider the matrix A = [1 0 0; 0 1 0; 0 0 0]. The column space of this matrix is the xy-plane, while the null space is the z-axis. Therefore, Col(A) is a proper subset of Nul(A).

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