Linear Algebra: Null Spaces

Question

Problem Statement: One important feature of the Gramian A^T×A of a matrix A is that is has the same null space of A (i.e. Null(A^T×A) = Null(A)). Let A be an m by n matrix.

a) Suppose v ∊ Null(A). Show that v ∊ Null(A^T×A).

b) Suppose v ∊ Null(A^T×A). Show that v ∊ Null(A) by showing that ||Av||^2=0

I'm not sure how to go about solving this problem. I would really appreciate some assistance. Thank you!

Answer 1

Part (a):

Show that if v∈ Null(A), then v∈Null(A^TA).

Definition of Null Space: By definition, if v∈Null(A), this means that:

Av=0.

Applying A^TA :

We need to show that v∈Null(A^TA). This means we need to show that:

A^TAv=0.

Substituting Av=0:

Since Av=0, we can write:

A^TAv=A^T(Av)=A^T(0)=0.

Thus, we have shown that:

A^TAv=0 ⟹ v∈Null(A^TA).

Part (b): Show that if v∈Null(A^TA), then v∈Null(A).

Definition of Null Space: If v∈Null(A^TA), then A^TAv=0.

Taking the Inner Product we can rewrite this as:

∥Av∥²=(Av)^T(Av)=0.

Here, ∥Av∥² represents the squared norm of the vector Av.

The squared norm ∥Av∥²=0 implies that Av=0.