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Say I have vectors $x, y$, then is $\text{proj }_x y $ a scalar multiple of $x$?

I have a book saying that it is, but I have no clue why this true. Is this really true?

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    What does the book say about it? Please share it. – 2017-02-06

3 Answers 3

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By definition, $\textrm{proj}_xy\in\textrm{span}(x)$, so yes, this is true.

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Of course. What else could the projection be? It has to be an element of the subspace spanned by $y$, which consists precisely of the scalar multiples of $x$.

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Yes this is true. You should check any definitions/formulas for the projection that you have.

In general, $$\operatorname{proj}_x y = \frac{\langle y,x\rangle }{\langle x,x\rangle } x$$ from which your claim is clear.