If Ax = 0 for every x in $R^n$, A is a zero matrix

Question

If A is a $m$ $x$ $n$ matrix such that: $Ax = 0$, for every $x \in \mathbb{R}^n$, then A = 0 is the zero matrix.

I have no idea how to begin proving this. The only hint is to consider a $j$ column in the Identity matrix $I_n$, but I don't see how that helps. Can anyone help me?

Hint: What do you get if you multiply a matrix with a vector that contains only zeros except one 1? — sranthrop, Oct 15 '15 at 23:34
@sranthrop, don't you just get a single vector, and its value is the x in the $ij^{th}$ position (where $I_{ij} = 1$)? — q.Then, Oct 15 '15 at 23:37

score 3 · Answer 1 · answered Oct 15 '15 at 23:35

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The $j$th column of the matrix $A$ is the image of the $j$th coordinate vector under $A$. Now what does the question tell us about these images?

answered Oct 15 '15 at 23:35

Daron

10,300

score 0 · Accepted Answer · answered Oct 15 '15 at 23:34

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To see what goes on, take an arbitrary non-zero $2\times 2$ matrix. Assume for a second all its entries are non-zero. Can you think of a single $x\in \mathbb R^2$ with $Ax\ne 0$? Now, what happens if one or more of the entries (but not all!) are zero? what can you do? Now what happens if it's an arbitrary matrix?

answered Oct 15 '15 at 23:34

Ittay Weiss

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I can see why A must be the zero matrix when you explained it like this, but how do I give a proof of showing concretely that A is the zero matrix? – q.Then Oct 15 '15 at 23:39
Let $A\ne 0$ be an arbitrary metric satisfying the stated property. We may assume the entry $a_{ij}\ne 0$. But then ... contradicting the given property of the matrix. – Ittay Weiss Oct 15 '15 at 23:41

If Ax = 0 for every x in $R^n$, A is a zero matrix

2 Answers2