# Orthogonal complement (linear algebra)

1. The problem statement

let $$\vec x$$ and $$\vec y$$ be linearly independent vectors in $$R^n$$ and let $$S=\text{span}(\vect x, \vect y).$$ Define the matrix $$A$$ as

$$A=\vec x \vec y^T + \vec y \vec x^T$$.​

Show that $$N(A)=S^{\bot}$$.

2.equations
I have a theorem that says$$N(A) = R(A^T)^{\bot}$$.
$$A$$ is symmetric; $$A = A^T$$.

3.Plan of attack
From the given above, it follows that if i can proof that $$S$$ is the orthogonal complement of $$R(A)$$, i'll be done. To do that, i'll have to show that all elements of $$S$$ are orthogonal to $$R(A)$$, and that any vector orthogonal to $$R(A)$$ is part of $$S$$.

Thus i want to show that
(I)the vectors $$\vec x$$ and $$\vec y$$ are orthogonal to any vector $$\vec z \in R(A)$$
(II)any vector $$\vec k$$ that is orthogonal to all vectors $$\vec z \in R(A)$$ can be written as a linear combination of $$\vec x$$ and $$\vec y$$.

I'm really not seeing how to do this. Hope someone can help me out, or at least tell me if i'm on the right track. Cheers.

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## Answers and Replies

vela
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I think it would be easier just to show N(A)⊂S and S⊂N(A). Proving both directions is pretty straightforward.