Exercise 5.5.3

Question

Let $A$ be an $m 	imes n$ matrix. Show that Ker $A$ = Ker $(A^*A)$.

Jing Huang · Accepted Answer

It is easy to see Ker $A \subset $ Ker $(A^*A)$. Next we show Ker $(A^*A) \subset $ Ker $A$. Consider $||A \mathbf{x}||^2 = (A \mathbf{x}, A \mathbf{x}) = \mathbf{x}^* A^* A \mathbf{x}$. Thus if $A^* A \mathbf{x = 0}$, we have $\mathbf{x}^* A^* A \mathbf{x} = ||A \mathbf{x}||^2 = 0$, i.e., $A \mathbf{x = 0}$. Thus Ker $(A^*A) \subset $ Ker $A$. As a result, we can conclude Ker $A = $ Ker $(A^*A)$.

Exercise 5.5.3

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Exercise 5.5.3

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