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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26 views

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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26 views

Matrix of Isometry

From Axler's Linear Algebra Done Right, $S$ is an isometry is equivalent to there being an orthonormal basis of $V$ such that $S$ has a block diagonal matrix where each block matrix is either$[1]$ or $...
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45 views

Show that if the vectors $v_1,v_2,\ldots,v_n\in \mathbb{R}^d$ are linearly independent, then…

Show that if the vectors $v_1,v_2,\ldots,v_n\in \mathbb{R}^d$ are linearly independent, then different linear combinations of them give different vectors. I'm guessing the question is asking for $a,b\...
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1answer
40 views

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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23 views

Block matrix multiplication example

Let \begin{equation} 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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7 views

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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1answer
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Find the matrix representation of $A: \mathbb R^4 \to \mathbb R^4$ in the standard basis [closed]

Given $$\begin{matrix} a_1=(1,1,-1,1) & a_2=(1,-1,1,0) \\ b_1=(1,0,1,-1) & b_2=(-1,1,1,1) \end{matrix}$$ Let $A$ be a linear operator in $\mathbb R^4$ such that $$\operatorname{ker}(A)=\...
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1answer
34 views

Show that if vector $\overline{v}$ is already at projection subspace $U$ that $\text{proj}_u(\overline{v})=\overline{v}$

Problem Let's assume that $V$ is finite-dimensional inner product space and $U$ is its subspace. Also, assume that $v \in V$. Show that $\overline{v} \in U \iff \text{proj}_U(\overline{v})=\overline{...
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5answers
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Linear algebra - Uniqueness of vectors

Maybe this question does not make much sense, but I have not seen a clear explanation of this so far. Let $V$ be a vector space. In abstract terms, what makes one vector $v_1$ different from another ...
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1answer
37 views

Explain whether “unordered” bases and “distinct” bases of a vector space meaning same thing? [duplicate]

Is the words unordered bases and distinct bases of a vector space meaning same thing ? Actually I have to solve the following problem. $\underline{Problem}:$ Find the number of distinct ...
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1answer
43 views

Gradient of $X^{T}AX$ wrt matrix X

I need to calculate the gradient of $X^{T}AX$ with respect to $X$, where $X$ and $A$ are $nxn$ matrixes. Using the matrix cookboox here (pag 9), I see that the gradient is obtained for each element $...
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2answers
44 views

How can I find the rank of this linear transformation

Suppose $Q \in M_{3 \times 3}\mathbb(R)$ is a matrix of rank $2$. Let $T : M_{3 \times 3}\mathbb(R) \to M_{3 \times 3}\mathbb(R)$ be the linear transformation defined by $T(P) = PQ$. Then rank of T ...
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2answers
83 views

Prove that this determinant equals zero

I recently came up with a problem in which I need to use the fact that the determinant below is equal to zero : \begin{vmatrix} 0 & a_{21} & 0 & a_{41} & 0 & \dots&0 &a_{...
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3answers
43 views

Change of basis changes rank of matrix

I have $3\times 3$ matrix $A$ with full rank in basis $\mathcal{B}_1$. If I change the basis to $\mathcal{B}_2$, I get matrix $A'$. But does the rank of $A'$ changes or stays $3$. I would say it ...
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2answers
92 views

Do the eight axioms of vector space imply closure?

This post is similar to my question but I do not quite understand the explanation. Usually, when we try to prove whether a set is a vector space, we will check closure on summation and ...
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3answers
67 views

How to calculate $\det(I +A^{50})$ with eigenvalues given

Let $A$ be a $3 *3$ matrix with the eigenvalues $1,-1,0$. How to calculate $\det(I +A^{50})$ ? I know that the answer is 4, but i have no idea how to approach such a problem
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0answers
15 views

Canonical injection

Can we say canonical injection is equal to the natural injection? I actually can not find an exact definition for canonical. For example if $ F_M : M \rightarrow M'$ and $F_X : X \rightarrow X' $ ...
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2answers
52 views

Understand the rank of $ \begin{bmatrix} A&b\\ b^{*}&0 \end{bmatrix}$

Let $A \in M_n(C)$ and $b$ be a column vector of n complex complements. Denote $\widetilde A = \begin{bmatrix} A&b\\ b^{*}&0 \end{bmatrix} $ If $rank(\widetilde A)=rank(A)$, which of the ...
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1answer
37 views

To prove a property of a positive semidefinite matrix with the zero first entry.

I want to prove the following: If the matrix $$ M=\begin{pmatrix} 0&\vec{q}^T \\ \vec{q}&N \end{pmatrix} $$ is PSD, then $\vec{q}=\vec{0}$. The only three properties of a positive ...
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1answer
38 views

Characterising the cone of elements whose inner product with $v \otimes v$ is non-negative

Let $V$ be an $n$-dimensional real inner product space. Consider the space $V \otimes V$, endowed with the tensor product metric, i.e. $$ \langle v_1 \otimes v_2 , w_1 \otimes w_2\rangle := \langle ...
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0answers
38 views

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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1answer
11 views

start T in matrix equation

In many start equations I see T (transpose) or -1 (inverse). Why is there using T, but not original matrix? Example https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula ($\mathbf{A}$ + $\...
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2answers
41 views

Similarity transformation into symmetric matrix

I have a matrix of the form: $$ \begin{bmatrix} 0 & q & 0 & 0 & 0 & 0 & \cdots \\ p & 0 & q & 0 & 0 & 0 &...
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0answers
26 views

The solution space of $A_{12}X=0$ and $B_{12}X=0$ is isomorphic?

Let $A$ be a $n\times n$ invertible matrix. Suppose $$A=\begin{pmatrix}A_{11}&A_{12}\\ A_{21}&A_{22} \end{pmatrix}$$ $$A^{-1}=\begin{pmatrix}B_{11}&B_{12}\\ B_{21}&B_{22} \end{pmatrix}...
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1answer
25 views

Linear transformation diagonalization with unknown vectors in basis

I've been working for some hours with this problem but I still can't get it. The problem says as follows: Given $ B=\{V_1, V_2, V_3\} $ and $ B'= \{V_1, V_1+V_2,-V_1-2V_2-V_3\} $ , basis of a vector ...
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2answers
71 views

Do real eigenvalues $\implies$ symmetric matrix? And why is a positive definite matrix symmetric?

Proof: $Av$ = $\lambda v$ $\implies \bar{v}^{T}Av = \lambda \bar{v}^{T} v$ ------(1) And, $Av$ = $\lambda v \implies \bar{A}\bar{v}$=$\bar{\lambda}\bar{v} \implies \bar{v}^{T}\bar{A}^{T}=\bar{\...
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1answer
37 views

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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1answer
51 views

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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1answer
36 views

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 ...
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0answers
23 views

Prove that if a linear system has infinitely many solutions, then any solution could be written as a linear function of free parameters, thanks.

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 ...
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3answers
61 views

$V=\{A\in M_n(\mathbb Q): \operatorname{tr}A=0\}$, Prove that $V\oplus \operatorname{Span}\{I_n\}=M_n(\mathbb Q)$

Q: Let $V=\{A\in M_n(\mathbb Q): \operatorname{tr}A=0\}$. Prove that $V\oplus \operatorname{Span}\{I_n\}=M_n(\mathbb Q)$ Since $\dim(V\cap \operatorname{Span}\{I_n\})=0$, $\dim(\operatorname{Span}\{...
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0answers
33 views

A polynomial basis of Cn[X]

Let $\mathrm{c}_{0}, \ldots, c_{n}$ be pairwise disctinct complexes. I proved that $\left(\left(X-c_{i}\right)^{n}\right)_{0 \leq i \leq n}$ is a basis of $\mathbb{C}_{n}[X]$ by induction. 1) Do you ...
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1answer
52 views

Why are self-adjoint operators important?

I am learning about self-adjoint and normal operators. So far, they have come up in the Spectral theorem, which says self-adjoint operators have an eigenvalue basis and a corresponding diagonal ...
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2answers
57 views

Inductively simplify specific Vandermonde determinant

From Serge Lang's Linear Algebra: Let $x_1$, $x_2$, $x_3$ be numbers. Show that: $$\begin{vmatrix} 1 & x_1 & x_1^2\\ 1 &x_2 & x_2^2\\ 1 & x_3 & x_3^2 \end{vmatrix}=(...
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1answer
44 views

Prove: if $W$ is subspace of $V$ and vector $v \in W$, then orthogonal projection of vector $v$ onto $W$ is $v$ itself

Let's assume that $\bar v \in V$. Let's also assume that $\bar v \in W$, where $W$ is a subspace of $V$. How to prove that then orthogonal projection $\operatorname{proj}_{W} (\bar v) = {\left\...
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0answers
27 views

Problem on quadratic form

Show that $\{x : x^{T}Ax \le 1\}$ is bounded if $A$ is positive definite, where $x^T=[x_1,x_2,...,x_n]$ and $A$ is an $n\times n$ matrix.
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2answers
65 views

when is $AB = B^TA$ true?

In what particular situation would the following be true? $AB = B^TA$ where $A$ is symmetrical, $B$ is not. I also know that $BB = B$.
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0answers
60 views

Writing the dual of a linear minimisation problem

I am trying to write down the dual of a linear minimisation problem and I would like your help to double check whether I'm doing it right. I'm following the instruction here . The original ...
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1answer
55 views

Generalizing the conjugate gradient like this works?

Given $A \in \mathbb{R}^{n \times n}$, a SPD matrix, and a vector $b \in \mathbb{R}^n$, it is possible to solve the problem $$\min_x \| Ax - b\|$$ with the conjugate gradient method. Its algorithm ...
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1answer
26 views

Let $U$ be a subspace of $V$, show that $(U^{0})_0 = U$

It is pretty obvious why $(U^{0})_0 \supseteq U$. but how do I show that $(U^{0})_0 \subseteq U$? For clarification: $U^{0} = \left \{ \phi\in V^{*} \mid \phi(a) = 0, for \; u \in U\right \}$ $(...
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1answer
33 views

If $V$ is a vector space, $W$ and $U$ are subspaces of $V$, why is $U^0, W^0 \supseteq ( U \cap W)^0$ true?

If $\textsf V$ is a vector space, $\textsf W$ and $\textsf U$ are subspaces of $\textsf V$, why is $\textsf U^0, \textsf W^0 \supseteq (\textsf U \cap \textsf W)^0$ true? My mind tells me that the ...
2
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1answer
22 views

Dimension of the space of matrices which is commutative to a given matrix.

Suppose I have a matrix $A $ in the space $ V $ of $n $ by $n $ matrices. Then it is quite clear that $S=\{B : AB=BA\} $ form a subspace. I want to find out its dimension. I think it depends on the ...
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0answers
34 views

Proof that equation $Lf=u$ guarantees satisfiability of weak form $\langle Lf,v \rangle = \langle u ,v \rangle$

Problem Let's observe in a closed interval $[a,b] \subset \mathbb{R}$ real-valued and continuous vectorspace $\mathcal{F}([a,b],\mathbb{R})$. Where $\langle ., .\rangle$ is some scalar product. This ...
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1answer
53 views

A quadratic in two variables can be factored if the determinant is $0$

(All real numbers) Show that $$ax^2+2hxy+by^2+2gx+2fy+c=0$$ can be factored as $$(a_1x+b_1y+c_1)(a_2x+b_2y+c_2)=0$$ iff $$\begin{vmatrix} a&h&g\\ h&b&f\\g&f&c \end{vmatrix} = ...
1
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1answer
21 views

Linearly dependent matrix columns implies zeroes in the solution of the system of equation given by this matrix?

Let's assume that we have a system of equations: $a_{11} x_1 + ... + a_{1n} x_n = 0$ $a_{21} x_1 + ... + a_{2n} x_n = 0$ $......................$ $a_{m1} x_1 + ... + a_{mn} x_n = 0$ Let us denote by ...
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0answers
31 views

Doubt in Hoffman and Kunze Section 5.2 (existence of determinant)

I am trying to read Hoffman Kunze's book on linear algebra and I have a doubt in a particular result, (Theorem 1) of Section 5.2. Specifically, the theorem states: Let $n > 1$ and let $D$ be an ...
2
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0answers
25 views

Different approaches to calculating angle between two vectors

I can calculate angle θ between two dimensional vectors a and b as the inverse cosine of their dot product divided by their magnitude: ...
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1answer
32 views

Product of two matrices in block form: step in proof

Let $R$ be a commutative ring with $1$ and $M_{k,l}(R)$ denote set of matrices of size $k\times l$ over $R$. $A\in M_{m,n}(R)$ and $B\in M_{n,p}(R)$. Partition $m$, $n$ and $p$ as $$ m=m_1+\cdots + ...