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Suppose a vector $a$ is given. What is precisely meant by projecting the component of vector $b$ out of $a$? Does that mean that the resulting vector $a_1$ (obtained by "projecting out the component of $b$ from $a$") is orthogonal to $b$? How could one achieve this?

I read that Gram-Schmidt process is used for this; however, it involves the notion of "normalization", and I'm not sure if that means that the vector $a_1$ would be normalized.

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It means if you decompose $b$ into a vector $b_1$ parallel to $a$ and a vector $b_2$ perpendicular to $a$, $b_1$ would be the answer. Gram Schmidt (GS) is used to find $b_2$, which is normal to $a$, and then $b_1$ can be found by subtracting.


The purpose (context) for which GS is introduced in most textbooks is to take a number of linearly independent vector (a basis for some vector space) and transform them into an orthogonal basis for the same space. In the case of two vectors above, you go from the basis $(a, b)$ to the basis $(a, b_2)$. In the case of a three-basis $(a, b, c)$ you first find $b_2$, so that you have $(a, b_2, c)$. Then you apply GS on $c$ to find a vector $c_2$ normal to the space given by $(a, b_2)$, but so that $(a, b_2, c_2)$ is the same vector space as $(a, b, c)$

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    But does that mean that $a_1$ has component of $b$ "projected out"? I'm mainly concerned with this term "projected out".2012-08-14