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Relationship between projection of $y$ onto $x_1, x_2$ individually vs. projection on both?

Mathematics Asked by roulette01 on September 29, 2020

This is essentially similar to the question I just asked on cross validated, but here I am going to pose it in a linear algebra way.

Consider $y in mathbb{R}^n$ and $x_1, x_2, 1_n in mathbb{R}^{n}$. Suppose you orthogonally project $y$ onto $x_1, 1_n$ and find the projection of $y$ onto the subspace spanned by $x_1, 1_n$ can be written as $hat{y}_1 = hat{beta}_1 x_1 + b_1$, i.e., a linear combination of $x_1$ plus some offset. Now do the same for orthogonal projection of $y$ onto $x_2, 1_n$ and find $hat{y}_2 = hat{beta}_2 x_2 + b_2$.

Now consider projecting $y$ onto the subspace spanned by both $x_1, x_2, 1_n$ and find $hat{y}_{12} = hat{gamma}_1 x_1 + hat{gamma}_2x_2 + b_{12}$.

If $x_1 perp x_2$, then I know $hat{beta}_i = hat{gamma}_i$. But what if they’re not orthogonal?

What can I say about the relationship between $hat{beta}$ and $hat{gamma}$ in this case?

Some specific questions that I am also interested in is if $hat{beta} >0 $, does this imply $hat{gamma} > 0$? If $x_1, x_2$ are linearly dependent, then I don’t think this won’t be true for one of the coefficients.

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