I have trouble understanding the topic of projection vs. least square approximation in an Introductory Linear Algebra class. I know this question has already been asked (Difference between orthogonal projection and least squares solution), but I want to check my understanding. PROJECTION ONTO SUBSPACE In projection, the purpose is to find the point where the […]

In an $n$-dimensional inner product space $V$, I have $k$ ($k\le n$) linearly independent vectors $\{b_1,b_2,\cdots,b_k\}$ spanning a subspace $U$. The $k$ vectors need not be orthogonal. Then I was told that the projection of an arbitrary vector $c$ onto $U$ is given by $$P_Uc=A(A^TA)^{-1}A^Tc$$ where $A$ is the $n\times k$ matrix with column vectors […]

I am looking for an intuitive reason for a projection matrix of an orthogonal projection to be symmetric. The algebraic proof is straightforward yet somewhat unsatisfactory. Take for example another property: $P=P^2$. It’s clear that applying the projection one more time shouldn’t change anything and hence the equality. So what’s the reason behind $P^T=P$?

If I have a projection $T$ on a finite dimensional vector space $V$, how do I show that $T$ is diagonalizable?

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