# Tagged Questions

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### Intuition/Picture - Matrix Multiplication - Product of [Row or Column Vector] and Matrix [Lay P95]

This question is not a duplicate of the original, in which user Shuchang proved the question. Presently I'm asking about further intuition or a picture, and no proofs please. $1.$ Intuitively, in ...
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### What's the fastest way to determine Eigenvalues & Eigenvectors of any 2 by 2 Matrix?

My instructor claims that it's inefficient and superfluous to compute eigenvectors de novo for each $2$ by $2$ matrix. He suggested a trick instead which resembles the eigenvectors and cases here. ...
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### rank($A$)=rank($A^T$) [duplicate]

Is there an elementary explanation of why the row-rank of a matrix equals its column-rank (without using adjoint maps, resp. lots of technical computations)? What is the geometric intuition behind ...
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### Intuition and Motivation - Linear Operator $T - \lambda_k I$ ? [Lay P270 Thm 5.1.2]

Let $T$ be a linear operator on a vector space V, and let $\lambda_{1},\ \lambda_{2},\ \ldots,\ \lambda_{k}$ be distinct eigenvalues of T. If $v_{1},\ v_{2},\ \ldots,\ v_{k}$ are eigenvectors of $T$ ...
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### Gaining Linear Algebra Intuition — Subspaces

So I aced linear algebra over the fall semester, though I'm deeply troubled in that I struggle to really describe what I did. I cannot say with confidence what it all meant, nor do I have any sort of ...
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### Geometric intuition behind subspaces in $\mathbb C^n$

While learning elementary linear algebra one develops a great deal of geometric intuition in $\mathbb R^n$. It helps to see the forest for the trees and leads through proofs. After meeting ...
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### Cauchy-Schwarz Inequality - Proof using Projections [Lay P379 Thm 6.7.16]

t If $u=0$, then the inequality becomes $0 \le 0$, which is true. See Practice Problem 6.7.1 on P382. If $u\neq 0$, let $W$ be the subspace spanned by $u$. $1.$ How would one determine to ...
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### Intuition - If $Ax = b$ has infinitely many solutions, why can't $Ax = c$ have only one solution? [Strang P165 3.4. 22]

If $\mathbf{Ax = b}$ has infinitely many solutions, why is it impossible for $\mathbf{Ax = c}$ (where $\mathbf{c}$ is a new right side) to have only one solution? Proof : Take two solutions of ...