Prove that if $y=(y_1, \ldots, y_n)$ is such that $y_1a_1 + \cdots + y_na_n = 0$, then $∀x ∈ \mathbb{R}^n$, $Ax · y = 0$

Question

I have no idea how to start the following question. Any help will be greatly appreciated.

(a) Let $A$ be a $n\times n$ matrix and let $a_1,\ldots,a_n$ be the rows of $A.$ Suppose $y=(y_1, \ldots, y_n)$ is such that $y_1a_1 + \cdots + y_na_n = 0$. Prove that, $∀x ∈ \mathbb{R}^n$, $Ax · y = 0.$

(b) For $n≥2$, find a nonzero $n×n$ matrix $A$ such that $∀x ∈ \mathbb{R}^n, Ax·x=0.$

Answer 1

Here's a hint for the first one. Write the linear combination $y_1 a_1 + \dots+y_n a_n =A^Ty$. Once you see how this helps the first, you should see how thinking about $A$ and $A^T$ solves the second.

Answer 2

(a) The condition on $y$ means exactly that $$y^TA=0.$$ i.e. we find the null row vector when we left multiply $A$ by the row vector $y^T$. Note that by $y^T$ I mean the row vector written horizontally when involved in matrix multiplication, while $y$ would be the column vector written vertically.

Now recall that by definition, the inner product you allude to is $$v\cdot w=\sum_{j=1}^nv_jw_j=v^Tw=w^Tv$$ where the right tow terms denote matrix multiplication (more precisely, in this case, a row vector on the left times a column vector with same size on the right, hence a real number).

Hence for every $x$, we have $$Ax\cdot y=y^T(Ax)=(y^TA)x=0x=0.$$

(b) Here is a $2\times 2$ example $$ \left(\matrix{0&1\\-1&0}\right) $$ Extend by $0$ to get an $n\times n$ example for every $n\geq 2$.