Questions tagged [linear-algebra]

For questions about 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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For which alpha does this system of equations spiral towards the orgin?

I am stuck with a math problem and I hope that somebody could help me in the right direction. Question: For which values of $\alpha$ do the trajectories of the solutions of the systems $x^{\prime}=A x$...
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Basis of polynomial vector space with conditions

I understand that the monomial basis proposed in this answer: $\{1,x,x^2,x^3,\ldots,x^n\}$ spans a regular polynomial vector space, but what process would I use to create a basis when there is ...
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How do i generate cubic coordinate-space traversal functions?

Goal- generation of a function f(x) such that for each x = 0 to infinite, f(x) outputs a point in a coordinate space ...
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Why do two distinct hyperplanes in $\mathbb{R}^4$ intersect in a plane?

Any tips? Ideas? A proof would be helpful
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Proof verification for $\rm dim range \,ST \le \min (dim range\, S, dim range\, T)$.

$U,V$ are finite dimensional vector spaces. $W$ is a given vector space. Let $T\in L(U,V), S\in L(V,W)$ be linear transformations then it is to be shown that $\rm dimrange \,ST\le\min (dimrange\, T, ...
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A link between multiplicative function on endomorphisms and their determinants

Take $E$ a $K$-vector space with $K$ a field with characteristic $0$. Then take $$ F:\operatorname{End}_K(E) \to K $$ a map such that $$F(1)=1$$ and $$ F(\phi)F(\psi) = F(\phi)F(\psi) $$ Then there ...
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Linear Transformation mapping linearly independent vectors onto a linearly dependent set

Let $T:\mathbb{R}^n \rightarrow \mathbb{R}^m$ be a linear transformation. Show that if $T$ maps two linearly independent vectors onto a linearly dependent set, then the equation $T($x$)=$0 has a ...
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2 votes
1 answer
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Vector Field With Elliptical Form?

I was doing some research when I found out this vector field, and I couldn't find its equations nowhere. How could I get the equation that describes this vector field, using only vector calculus and ...
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Question about rref and basis

I already asked this but didn't really get the answer but my question is basically if I have a matrix A, from what I've understood there are a couple ways to interpet a matrix like as a collection of ...
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4 votes
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Is $\text{GL}_2(\mathbb{R})/\mathbb{R}^{\times}$ isomorphic to $\text{SL}_2(\mathbb{R})$?

Let $\text{GL}_2(\mathbb{R})$ be the set of real $2\times 2$ invertible matrices (where the operation is matrix multiplication). It has $\text{SL}_2(\mathbb{R})=\{ A \in \text{GL}_2(\mathbb{R}| \det A ...
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Prove Cauchy-Schwarz $\langle x,y\rangle \leq \|x\|\ \|y\|$ by letting $ c= \dfrac{1}{\|x\|}, d = \dfrac{1}{\|y\|}$

Suppose $V$ is a inner product space and $x,y\ne 0$, prove Cauchy-Schwarz $$\langle x,y\rangle \leq \|x\|\ \|y\|$$ by leting $$ c= \dfrac{1}{\|x\|}, d = \dfrac{1}{\|y\|}$$ and the fact that $$ \|cx\pm ...
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Proof for Cauchy Schwarz using the trig definition of the dot product

The Cauchy Schwarz inequality for the dot product of $n$ dimensional vectors states: $$|\textbf{u}\cdot \textbf{v}|^2\leq|\textbf{u}|^2|\textbf{v}|^2 $$ The Wikipedia proof at point 4.4.1 of makes ...
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Reverse a mod in an equation

Being quick and to the point, I have the following equation. (A * M)%F = B I want to solve for M. How do you move the modular F?
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how do i solve non linear equation? TOPIC IS PDE [closed]

Find a separated solution of the following nonlinear wave equation: ∂u/∂t=cu ∂y/∂x and What is a separated solution of the 2 -dimensional wave equation (∂^2 u)/(∂t^2 )=a (∂^2 u)/(∂x^2 )+b (∂^2 u)/(∂y^...
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Finding the maximal direct sum decomposition of a linear map

Suppose a matrix $M$ over some field $k$ allows a diagonalisation $P D P^{-1}$. On the level of linear maps, this is a decomposition of the linear map $f: V \to V$ represented by $M$ into $$V \cong \...
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Show that $\operatorname{max}\{x^TAx: x \in \mathbb{R}^n, x\geq 0, ||x||_2=1\} \leq \operatorname{max}\{x^THx: x\geq 0, ||x||_2=1\}$

Let $A\in M_n(\mathbb{R})$ be nonnegative and consider the symmetric nonnegative matrix $H=\frac{1}{2}(A+A^T)$. I want to show that $$\operatorname{max}\{x^TAx: x \in \mathbb{R}^n, x\geq 0, ||x||_2=1\}...
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Tensor Product, proof of isomorphism

Let V and W be two vector spaces and $$\Phi: Hom_K(\land^rV,W)\rightarrow Alt_K^r(V,W)\\ f \mapsto [(v_1,....,v_r)\mapsto f(v_1\land \cdot \cdot \cdot \land v_r)]$$ I need to show that $\Phi$ is an ...
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Inner product of operators [closed]

Suppose $A, \ B \in \mathcal{L}(\mathbb{C^n}) $, Where $\mathcal{L}(\mathbb{C}^n)$ is the space of linear operators acting on the $n$-dimensional complex vector space $\mathbb{C}^n$. $\mathcal{L(\...
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Linear Algebra - Tensor Products

Let $V$ be a finite-dimensional vector space over $\Bbb Q$ and $\bigwedge^kV$ be the $k$-th antisymmetric power. Let $\{ v_1, \dots, v_n \}$ be a basis of V. Define $$\pi: V^{\otimes k} \to \bigwedge^...
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Lower bound on the off-diagonal elements of a PSD matrix

Suppose we have a PSD matrix $X\in\mathbb{R}^{2d}$, which could be written in the following block form $$X=[X_1\quad X_2;\quad X_2^\top\quad X_3],$$ where $X_1, X_3\in\mathbb{R}^d$ are PSD matrices, ...
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Linear Transformation from $\mathbb{R}^6$ to $P_2(\mathbb{R})$

I am trying to figure out what the transformation of the values {$x_1, x_2, x_3, x_4, x_5, x_6$} $\in \mathbb{R}^6$, becomes when transformed to $P_2(\mathbb{R})$. I understand that $P_2(\mathbb{R})$ ...
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Cauchy-Schwarz inequality and angle between two vectors

Notes I am reading these notes, and I can't understand the Cauchy-Schwarz inequality. It says that it proves that the input is between $[-1,1]$. The Cauchy-Schwarz inequality only states that the ...
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2 answers
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Multiplication of two random matrices over a finite field

Consider a matrix $\mathrm{X}$ sampled uniformly at random from the set of all rank $r$ matrices over $\mathbb{F}_q^{m \times n}$ and a matrix $\mathrm{Y}$ sampled uniformly at random from the set of ...
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1 answer
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Given a subspace $W$ of $V$ and a linearly independent set $A \subseteq W$, is $A$ linearly independent in $V$?

Can you please show that, if a set $A$ is linearly independent in $W$, a subspace of $V$, then it is linearly independent in $V$? Thank you. Here is my attempt. Let $A = \{ a_1, a_2, \cdots , a_n \}$ ...
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2 votes
1 answer
29 views

Let $B$ be simetric and positive definite. Show that $x^T\left(B - \frac{Bss^TB}{s^TBs}\right)x $ is also positive definite

Let $B\in \mathbb{R}^{n\times n}$ simetric and positive definite and $s\in\mathbb{R}^n-\{0\}$ Let $$M := B - \frac{Bss^TB}{s^TBs}$$ Show that $$x^TMx > 0$$ My try: $$x^TMx$$ This is $$x^T\left(B - \...
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Is the dot product of vectors $u$ and $v$ the same as the dot product of $u$ and $-v$?

Can’t find this online so asking it here: Is the dot product of vectors $u$ and $v$ the same as the dot product of $u$ and $-v$? I have to use this to solve a bigger problem and stuck on this part.
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Determine the value(s) of $a$ so that the following linear systems has a one solution, infinite solutions and no solution

System 1: \begin{align*} \begin{cases} x + 2y - 3z = 4\\ 3x - y + 5z = 2\\ 4x + y = -a^2 + a +16\\ \end{cases} \end{align*} System 2: \begin{align*} \begin{cases} ax + y + z = -2\\ x + ay + z = 1\\ x +...
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-2 votes
0 answers
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Math struggles for discrete math and Linear algebra [closed]

I am a second year student at a community college. I have autism and obsessed with my grades. I am struggling heavily when comes to these mathematics. I have tried youtube, my classmates, and my ...
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1 vote
1 answer
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Reference request - textbook on linear algebra

I'm looking for a textbook on linear algebra that is on a level higher than that of a first-read text, but isn't of the machine-gun-definition-theorem-proof-corollary type. I used Lay's text as an ...
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2 answers
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Find a pair of linear transformations that do not commute

The problem statement is as follows: Suppose $\mathbb{F}$ is any field. Find a pair of linear transformations $S,T \in \mathcal{L}(\mathbb{F^{2}}, \mathbb{F^2})$ such that $ST \neq TS$ My attempt ...
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2 votes
0 answers
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What does the bar symbol over a scalar means in the context of complex vector spaces?

In chapter 5.3 in "Elementary Linear Algebra" titled "Complex vector spaces" I came across a weird notation that I don't understand and can't find an example or explanation ...
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2 answers
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How to find the points that are in-between 4 planes

I have a set of points, I want to select only a few of those points. For that, I have 4 planes equations in the general form and I want to be able to check in a look if a given point would exist in ...
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1 vote
1 answer
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Where is my mistake in showing $\nabla \mathbf{a}^{T}\mathbf{x}=a_{1}+a_{2}+\cdots+a_{n}$?

Let $f:=\mathbf{a}^{T}\mathbf{x}$. The claim that: $$\nabla f=\nabla\mathbf{a}^{T}\mathbf{x}=\|\mathbf{a}\|_{1}=a_{1}+a_{2}+\cdots+a_{n}\tag{1}$$ is false where in fact the true answer is $\mathbf{a}^{...
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Second partial derivative: What happens if $f_{xy}^2$ is larger than $f_{xx} f_{yy}$

Considering $(a,b)$ is a critical point of a funciton $f(x,y)$ and $D(x,y) = det(H(f(x,y))) =f_{xx}(x,y)f_{yy}(x,y) - (f_{xy}(x,y))^{2}$ is the determinant of the hessian matrix from that function. If ...
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0 votes
1 answer
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How to prove that Quaternion's algebra over isomorphic to Mat2(Z [duplicate]

My ideas: I tried to build an explicit isomorphism, but as I think it is only possible when p = 1 (mod 4), and for p = 1 (mod 4) it get it. In my second attempt, I tried to look at them as vector ...
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2 votes
1 answer
24 views

Checking whether differential operator maps vector space to itself

Given a vector space $V$ that is spanned by: $\{3^t, 2^{-t}\}$, can we conclude that the differential operator $\frac{d}{dt}$ maps $V$ onto itself? It is my understanding that by looking at the basis ...
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How to visualise 3D vector rotation around a line?

If we want to rotate a 2D vector we need only angle $\theta$ by which we want to rotate the vector. And we have only two possibilities: one for clockwise rotation and other for counterclockwise ...
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1 vote
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Solving for cross-covariances between two random vectors

Let $\mathbf{x}$ and $\mathbf{y}$ be Gaussian random vectors with zero mean and covariances $\mathbf{C}_{\mathbf{xx}}$, $\mathbf{C}_{\mathbf{yy}}$, respectively. Define the sum of these processes as $\...
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Given two points, construct a parallelepiped with a square base(length given)

So I have been doing this for a while now, and I am afraid I am horribly overthinking this problem, so I came here for some fresh takes on this. I want to obtain all the vertices of a parallelepiped ...
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-2 votes
1 answer
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What is the formula for the orbital velocity of the Earth from x, y, z coordinates of ephemerides at set time intervals?

For example for step size 10080 minutes x y z v 1721057.5 B.C. 0001-Jan-01 -5.83E-01 7.93E-01 3.65E-03 1721064.5 B.C. 0001-Jan-08 -6.78E-01 7.16E-...
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1 vote
0 answers
57 views

Add some constraints to make $x^TAy \geq 0$ where $A \succ 0$

I am trying to prove the difference of two algorithms' asymptotic mean square error. Eventually, I get an expression like $x^TAy$, where $A$ is a positive definite matrix. I would love to see under ...
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1 vote
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Set functions: function of union of functions

I have a doubt about a property of set functions In detail, I have a set function $\omega$, which I apply to some subsets $S_1$, $S_2$, ... I noticed that this function has the following property: $\...
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The column is full rank,,does the matrix have a left inverse?

a non-square matrix$A_{m\times n}$,Its rank is equal to n. I know it may have a unique solution,maybe not. We take its left inverse to get the solution,But if its solution does not exist, does the ...
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Prove or disprove: $A \in M_{n,n}( \mathbb{C})$ is a self-adjoint matrix so $⟨\cdot,\cdot⟩_{A}$ positive definit is. It applies that $\det(A) > 0$.

$A \in M_{n,n} (\mathbb{C})$ is a self-adjoint matrix so that Hermitian form $\langle \cdot,\cdot\rangle_{A}$ positive definit is. It applies that $\det(A) > 0$. My idea is that the Hermitian form $...
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(Dis)prove for a matrix $A$ and for an unitary matrix $S$, where $D = S^{-1}AS$ a diagonal is which diagonalelements are imaginary that $A^{*} = - A$

$A \in M_{n,n}(\mathbb{C})$ a complex matrix and $ S \in U_{n}$ an unitary matrix with $D = S^{-1}AS$ a diagonalmatrix, which has only imaginary elements in it's diagonal. (the real part elements are ...
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  • 1,005
1 vote
1 answer
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Intersection of linear manifold

I have this question on my homework sheet that I don't know how to approach: "A three-dimensional linear manifold in R4 has the direction vectors a1=<2,0,1,0> a2=<1,1,0,0> a3=<0,1,...
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maximal eigenvalue of self-adjoint operator is non-degenerate

I want some help in this one, if someone can prove or disprove it: "If $T$ is a compact, self-adjoint operator with positive spectral gap, then $||T||_2$ is always an eigenvalue and the ...
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  • 101
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2 answers
58 views

Exercise 3.F.5 in "Linear Algebra Done Right 3rd Edition" by Sheldon Axler.

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. 3.94 Definition dual space, $V'$ The dual space of $V$, denoted $V'$, is the vector space of all linear functionals on ...
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-1 votes
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What is the relation between matrices $A^2$, $A^{-1}$, $A^2 + 2A + I$ to the diagonal matrix $D$ corresponding to $A$? [closed]

Find the eigenvalues and eigenvectors of $A^2$, $A^{-1}$, $A^2 + 2A + I$. Write the relation that connects these matrices to the diagonal matrix D corresponding to A. Can you use this relation to ...
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1 vote
1 answer
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Determinant of a linear map $\phi: \text{Mat}(n \times n, \mathbb{F}) \rightarrow \text{Mat}(n \times n, \mathbb{F})$

I have some difficulties applying the Kronecker-Delta to the elementary matrices, such that I can compute the determinant correctly. My problem is, that I have a linear map $\phi: \text{Mat}(n \times ...
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