# Questions tagged [linear-algebra]

Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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### Linear function $f$ and the existence of constant M depending of $f$

If $f: \mathbb{R}^m \rightarrow \mathbb{R}^n$ is a linear map, show that there exists a $M$, such that $|f(x)| \leq M|x|,$ for all $x \in \mathbb{R}^m.$ I guess $M$ depending of matrix associated ...
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### Linear independence proofs

Show that 1. One vector is linearly independent if and only if it is not the zero vector. Let $S = \{\mathbf{0},v_1, \ldots,v_n\}$ be a set of vectors. Then \begin{equation*} 1\times \mathbf{0}+0\...
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### Block matrix multiplication example

Let A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}, \qquad B = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 3 & 1 \\ 1 & 1 & 1 \end{pmatrix} \end{...
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### Given 2 points defining line segment, what are the points that define lines segments of equal size at distance h

On a 2d, euclidean space, given the line segment defined by points P=(p_x,p_y), and Q=(q_x,q_y), and height h, what are the points P_{+h},Q_{+h} and P_{-h},Q_{-h}, that define (two) line segments at ...
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### Prove of disprove.There exists 3x3 real valued matrix such that … [closed]

Prove of disprove this statement. There exists 3x3 real valued matrix B such that $B^2=A$. $$A=\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\1 & 1& 0 \\ \end{bmatrix}$$
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### Trilateration of a Received Signal

I'm trying to determine if this is even possible. I have 2 unknowns that I want to solve for. The strength of an RF pulse and it's location in xyz coordinates. The only knowns are the four locations ...
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### Comparing condition numbers

I have two matrices, $A$ and $B$ where $A,B \in R_{mxn}$ ($m>>n$). I need to answer the question: Which one is better-conditioned among $A$ and $B$ ? However, I will be making these comparisons ...
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### Question about points in general position (from Keith Conrad's expository paper on isometries)

https://kconrad.math.uconn.edu/blurbs/grouptheory/isometryRn.pdf In an expository paper on isometries of $\Bbb R^n$ Keith Conrad proves the following corollary: Corollary 2.7. Let $P_0,...,P_n$ be ...
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### Showing that there is unique matrix $B$ such that $B^k=A$ for some $A$

Let $A$ be a $n$ by $n$ real matrix with distinct positive eigenvalues $\lambda_1$,...,$\lambda_n$. And let $k$ be an odd integer. Then, I was able to show that there exists a real matrix $B$ such ...
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### For $4\times 3$ matrix $M$, for any $3\times4$ matrix $N$, $\exists 0 \neq v \in \mathbb{C}^4$ such that $MNv = 0$.

Define $M := \left(\begin{matrix}1&-1&2\\2&-1&1\\-4&1&0\\3&-2&3\end{matrix}\right)$. Prove or disprove: For every $3 \times 4$ complex matrix $N$, there is a non-zero ...
Suppose $Ax=b$, where $A$ is of dimension $q\times p$ with $q<p$, $rank(A)=q$, $b$ is $q \times 1$. Let $\mathcal{X}=\{x:Ax=b\}$ be the solution set of this system. How to rigorously prove that ...