The discussion centers around proving that the null space of a matrix \( A \), denoted \( N(A) \), is a subset of the null space of the product \( A^t A \). The context involves linear algebra concepts related to matrix properties and null spaces.
Some participants have provided insights into the definitions of null spaces and the conditions under which elements belong to these spaces. There is an ongoing exploration of the logical steps needed to establish the subset relationship, with participants questioning and clarifying the reasoning process.
Participants are working under the assumption that \( m < n \) and are discussing the implications of this condition on the null spaces involved. There is also a focus on the need for rigorous justification of claims made regarding the properties of the null spaces.
jbunniii said:If ##Ax = 0##, then what is ##A^t A x##?
If ##X## and ##Y## are sets, how do you prove that ##X \subset Y## in general?inter060708 said:A^t A x= 0
and how do I justify the subset part?
jbunniii said:If ##X## and ##Y## are sets, how do you prove that ##X \subset Y## in general?
Yes, that's right. So what is the defining property of an element of ##N(A)##? In other words, ##x \in N(A)## if and only if ...?inter060708 said:I need to show that elements in X also belongs to Y.
jbunniii said:Yes, that's right. So what is the defining property of an element of ##N(A)##? In other words, ##x \in N(A)## if and only if ...?
You have the right idea, but you need to state the logic correctly. The goal is to prove that if ##x \in N(A)## then ##x \in N(A^t A)##.inter060708 said:x ε N(A) iff Ax = 0 and since A^t A x = 0 then x ε N(A^t A).
Therefore x belongs to both N(A) and N(A^t A).
Is this correct?
inter060708 said:A^t A x= 0
jbunniii said:You have the right idea, but you need to state the logic correctly. The goal is to prove that if ##x \in N(A)## then ##x \in N(A^t A)##.
So, suppose ##x \in N(A)##. Then by definition, ##Ax = 0##. Therefore...?