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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Existence of a matrix similar to upper triangular matrix.

Suppose that $X \in M_{n\times n}(\mathbb R)$, then there exists a nonsingular $S \in M_{n \times n}(\mathbb R)$ s.t $$S^{-1}XS= \begin{pmatrix} P_{1} & & & \\ &P_{2} &\ &...
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Show that orthogonal projection of $(0,1,0)$ over $W$ is the vector $(\frac{1}{\sqrt{2}}, \frac{i}{2}, \frac{1+i}{4})$

Given $W = Sp\{(1,i,0),(1,2,1-i)\}$ a subspace of $\mathbb{C}^3$ with the standard inner product. Show that the orthogonal projection of $(0,1,0)$ over $W$ is the vector $(\frac{1}{\sqrt{2}}, \frac{i}{...
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A question about linear application

For example, we have two vector space V, W, and X is a K-base of V. So, therefore if we have $f,g: V \to W$ two K-linear applications. Suppose f(x)=g(x) for all $x \in X$, how to prove $f=g$? And for ...
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How to identify generator matrices for a linear code?

Lets say we have some linear code of length n over F2 that is given by: {a, b, c, d, e, f, g, h} where a, b, c, d, e, f, g, h represent some element/word in this code and each has length n. If I were ...
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Can there be a third type of unit other than 1 and i?

I know the question sounds vague, so let me try to explain what is on my mind. When we imagine the complex plane, one axis represents the real part and the other axis represents the imaginary part. ...
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1answer
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How to find out the magnitude of addition of three vectors?

There are three unit vectors. The question is to find the modulus of the addition of these three vectors: A=3i-2j+k B=2i-4j-3k C=-i+2j+2k
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Prove that exists an inner product such that $\{1+x,x+x^2,x^2+x^3,x^3+x^4,x^4-1\}$ is orthogonal

I've been asked to prove or disprove that exists an inner product in $\mathbb{R}_5[x]$ such that $A = \{1+x,x+x^2,x^2+x^3,x^3+x^4,x^4-1\}$ is orthogonal. How does one approach this?
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Question about a statement involving system of linear equations and rank of matrix.

the statement is: let $A\in \mathbb{R}^{m\times n}$, if for every $c\in \mathbb{R}^m$ there exists a solution for $Ax=c$, then $\operatorname{rank}(A)=m$. Now I can understand that this is a true ...
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Derivative of $vec(XX^T)$ (vectorized factor matrix)

I'm looking for the gradient of a factor structured matrix generated by $X X^T$, subject to some constraints, and I'm very thankful if someone could help me further. Specifically, I have a Lagrangian ...
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Show that $M$ and $D$ commute, where $J$ is a matrix in Jordan canonical form, $D$ is its diagonal entries and $M = J - D$.

Question: Be $J$ be a finite dimensional matrix in Jordan Canonical form, let $D$ be the diagonal matrix with diagonal entries equal to $J$, and let $M := J - D$ Show that $MD = DM$ Attempt: Let $(A)_{...
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Why does this converge to eigenvector?

I'm currently working on a way to analytically find eigenvectors. Teacher says to show that $ x_{k+1}\gets \dfrac{Ax_k}{||Ax_k||}$ where $x_{k}$ is the vector and $A$ the matrix converges to an ...
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Explaining the proof of the condition number in relation to the norms of the matrices.

We have a system of equations, $A \vec{x} = \vec{b}$, to which we introduce certain disturbances and get $A (\vec{x} + \delta \vec{x}) = (\vec{b} + \delta \vec{b})$. We can then subtract the second ...
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Right form for my Jacobian with only one function

I am faced to what seems to be simple : I have a function $\alpha_(b_{1,1},b_{2,1}) = \dfrac{b_{1,1}}{b_{2,1}}$. I mean that I have only one single function depending of 2 variables $b_{1,1}$ and $b_{...
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20 views

Prove equality of linear maps

Let $V,W$ be $K$-vector spaces and $S$ a $K$-basis of $V$. Suppose that $V$ is finite dimensional. Let $\varphi_1,\varphi_2: V → W$ be $K$-linear maps. Suppose $\varphi_1(s) = \varphi_2(s)$ for all $s ...
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1answer
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Finding an orthonormal basis z of $\Re^n$ with respect to which the matrix $A_z$ is diagonal

Let $\textbf{A}:\Re^n \rightarrow \Re^n$ be given by $\textbf{A} \vec{x}=\vec{x}+l(\vec{a},\vec{x})\vec{a}$ where $||\vec{a}|| = 1, \vec{a} \in \Re^n, l \in \Re.$ And let $\textbf{A}$ be orthogonal ...
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1answer
30 views

Factorization of a unimodular matrix

Let $S = \big(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \big)$ be a matrix with determinant $ad - bc =1$ (this is the case I'm interested in, but I don't think it is essential). ...
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73 views

Determinant of a sum of square matrices

Let $$A=\begin{bmatrix}0&1&0&\cdots 0\\ 0&0&1&\cdots 0\\ \vdots\\ 0&0&0&\cdots 1 \\ 1&1&1&\cdots1 \end{bmatrix}_{n\times n}$$ i.e. it has ones above ...
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1answer
17 views

$T$ injective equivalent to $T'$ surjective

I'm currently going through Sheldon Axler's "Linear Algebra Done Right" (3rd Ed.) and struggle with the result of 3.110, where he shows that $T\in\mathcal{L}(V, W)$ is injective if and only ...
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25 views

Existence of a non-zero T- invariant subspace on $\mathbb{Q}^{4}$

For any linear transformation $T:\mathbb{Q}^{4} \to \mathbb{Q}^{4} $, does there always exist a non-zero T- invariant subspace? As we know For any linear transformation $T:\mathbb{R}^{4} \to \mathbb{...
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Checking/testing bilinearity of a function

I have already aware of the bilinear form of a function which is: <x,y> =x^tAy= ∑_(i,j)=〖a_(i,j)x_iy_i〗 Applying this concept to an example: α(x,y)=x^T* [2 -1, *y -1 1] Assuming x = [x1 y1], ...
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41 views

If you have a set of eigenvectors with full multiplicity, is it always possible to orthogonalize them?

a) construct a (diagonalizable) 2×2 complex symmetric matrix not admitting an orthogonal basis of eigenvectors b) construct a 2×2 complex symmetric matrix which cannot be diagonalized The point of ...
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31 views

Dimension of Example Quotient Space

Let $V$ be the vector space across the set of all integers, with subspace $W$ as the vector space across all even numbers, both with the field as the integers. The quotient space $V/W=\{v+W|v\in{V}\}=\...
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Finding the determinant of unknown matrix

How can I solve this question: given that $A$ and $B$ are both $3\times 3$ matrices and $\det(A) = -44$ and $\det(B) = -2$. How can I find the determinant $\det(2A^{T}B^{-1})$?
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19 views

Using Cholesky Decomposition iteratively to find eigenvalues

I have an algorithm that should theoretically find eigenvalues of any symmetric, positive definite matrix $A$. Start with a seed $A_0 = A$ and then find its cholesky decomp, so that $A = U^T U$ then ...
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11 views

Analytic methods of computing entrywise absolute values of matrices

Are there any analytic ways of computing, or closely approximating the entry wise absolute value of a matrix? More explicitly, for an arbitrary matrix $B\in \mathbb{R}^{m\times n}$, are there any ...
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1answer
19 views

Is there any situation where the LDU decomposition is the same as the eigenvalue decomposition?

I was just wondering if there are any situation where the LDU decomposition is the same as eigenvalue decomposition (diagonalization)? The only way this can be possible if L and U are inverse so ...
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20 views

Set of all p-times differentiable functions is vectorspace

I'm stuck at the very basic and trivial task which is to prove that the set $M:=C^p(a,b)$ of all p-times differentiable functions $f: (a,b) \rightarrow \mathbb{R}$ form a vector space, whereas the set ...
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Geometry of vanishing lines

In page 220 of Multiple View Geometry in Computer Vision is the following figure: So in this figure, $\mathbf{l}$ is the vanishing line of the scene plane $\pi$ projected onto the image plane. What I ...
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2answers
33 views

Use Schur to prove that for any norm $\lim_{n \rightarrow \infty} ||A^n|| = 0 \text{ iff } \rho(A) < 1$ where $\rho$ denotes spectral radius.

I know I need to prove both directions of the biconditional, but am having some trouble in both ways. First, looking left to right, If we assume the limit equals zero, it means that the $A$ matrix ...
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Confusing Proof on Orthogonality and Least Squares

I'm reviewing this textbook proof and am struggling to understand the Least Squares proof that gives us the normal equations. They write: Suppose $\hat x$ satisfies $A\hat x = \hat b$. By the ...
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10 views

Unique decomposition of an affine bijection

Let $E$ be an affine space attached to a $K$-vector space $T$, $a\in E$ and $u:E\rightarrow E$ an affine bijection. There exists a unique affine bijection $u_1:E\rightarrow E$ such that $u_1(a)=a$ ...
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1answer
23 views

Distributive property of matrix-vector multiplication?

I know that matrix multiplication is not distributive but what about matrix-vector multiplication? If $A \in \mathbb{R^{m \times n}}$ and $\vec{x}+\delta \vec{x} \in \mathbb{R^{n \times 1}}$. Then can ...
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How Does One Use a Rotation Matrix to Calculate the Rotation About an Arbitrary Axis?

I have a rotation matrix for a body rotating about some arbitrary axis. I can determine the axis of rotation and the corresponding magnitude of the rotation. But I would like to know how to ...
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21 views

Bivectors and (2,0)-tensors

Given a vector space $ V $ of finite dimension $n$, you can consider his dual $ V^* $ i.e. the space of the linear functionals $ \omega:V\longrightarrow K $ (where $K$ in the field of the scalars), or,...
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The logistics of a V to W linear transformation [closed]

If T is a linear transformation from V to W, and w is not in Range(T), does that 2w is also not im Range(T)
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How to solve $x^TAy = d$, where $A$ and $d$ are known, x and y remain unknown.

How to solve $x^TAy = d$, where $A$ and $d$ are known, $x$ and $y$ remain unknown. $x$ and $y$ are vectors; $A$ is a matrix; $d$ is a scalar. There are constraints for $x$ and $y$, requiring $x$ and $...
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1answer
25 views

Consider the equation $2R*x^2 - (2R\cos\frac{B-C}{2})*x+r=0$

$2R*x^2 - (2R\cos\frac{B-C}{2})*x+r=0$ We have the triangle ABC Where $r = 4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}$ R is the circumradius of the triangle ABC. I have the following answer ...
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1answer
20 views

Functional Expectation

Problem: Let $X$ be distributed over the set $\Bbb N$ of non-negative integers, with pmf: $$P(X=i)=\frac{\alpha}{2^i}$$ $\alpha$ $E[X]$ For $Y=X$ mod $3$, find: $P(Y=1)$ $E[Y]$ I will assume $\...
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If $x_1,…,x_{n-1}$ make angle $\le\theta$ with some $n-1$ plane $P$, what is an upper bound for $\angle(P,\operatorname{span}\{x_1,…,x_{n-1}\})$?

It looks like the angle between $P$ and $\operatorname{span}\{x_1,\ldots,x_{n-1}\} := \tilde{P}$ should be at most $\theta$ or possibly some constant multiple. Here $\theta \in (0,\pi/2].$ Tried ...
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2answers
32 views

Linear independence of solutions to Floquet equation

I have a question about solutions to the Floquet equation, i.e.: $$ \dot{x} = A(t) x $$ where $x$ is an $N$-dimensional time-dependent column vector and $A(t)$ is a $T$-periodic $N \times N$ matrix. ...
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1answer
36 views

How to prove that the general solution of a homogeneous linear first order ODE has dimension 1?

I would like to prove it by using Linear Algebra. I first defined a Linear Application on an open interval $I$ $$L(.) : C^1(I) \to C(I).$$ $$y(x) \to y(x)'+p(x)y(x)$$ For a Homogeneous Equation $L(y)=...
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1answer
46 views

Difference between $p\left(x\right)\cdot{q\left(x\right)}$ and $\left(p\cdot{q}\right)\left(x\right)$

Let $p\left(x\right),q\left(x\right)\in{R\left[x\right]}$ such that $p\left(x\right)=x$ and $q\left(x\right)=x+a$. My lecture notes state that "$p\left(x\right)\cdot{q\left(x\right)}=x^2+xa$ and $...
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21 views

How to calculate determinant? [duplicate]

How to calculate the following determinant: $\begin{vmatrix} a & b & b & ... & b \\ b & a & b & ... & b \\ . & . & . & . ... & .\\ b & b &...
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1answer
19 views

Proving that the matrix representation of a nilpotent linear operator is upper-triangular with diagonal entries $ = 0$

Let T be a nilpotent operator on an $n$-dimensional vector space $V$, and suppose that $p$ is the smallest positive integer for which $T^P=T_0$ (zero-transformation). I have so far proved the ...
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1answer
19 views

Finding the eigenvalues and an orthornormal basis of eigenvectors of an orthogonal mapping

Let $\alpha = \{\vec{a_1},\vec{a_2},\vec{a_3},\vec{a_4}\}$ be an orthonormal basis for $\mathbf{R}^4$. Let $\mathbf{A}:\mathbf{R}^4 \rightarrow \mathbf{R}^4$ be a linear orthogonal map with the ...
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32 views

Nonlinear Numbers in Linear Algebra

I am writing an electronic circuit simulator, I know there are already tons of them out there but writing one yourself deepens your understanding tremendously. My approach that works for linear ...
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1answer
39 views

If two vectors are orthogonal then $\|a + b\|^2=\|a\|^2 + \|b\|^2$

Can we say that if $a$ and $b$ are orthogonal (meaning $\langle a,b \rangle = 0$) then $\|a + b\|^2=\|a\|^2 + \|b\|^2$ must be correct? Reasoning: $\|a+b\|^2 = \langle{a+b,a+b}\rangle \underset{...
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16 views

Morphism of complete collineation

A complete collineation of depth $d$ is the following data: $2(d+1)$ vector spaces $V_i$ and $W_i$ (of dimension d) and $d+1$ linear maps $\lambda_i:V_{i} \longrightarrow W_{i}$ for $i\in {0,...,d}$ ...
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18 views

$v_1,…,v_m,w$ linearly independent $\iff$ $w \notin \text{span}(v_1,…,v_m)$

Let $v_1,...,v_m$ linearly independant in $V$ and $w\in V$. Show that $v_1,...,v_m,w$ is linearly independent $\iff$ $w \notin \text{span}(v_1,...,v_m)$. It might be a "classic" result in ...
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Comparing GMRES to equivalent algorithms such as GCR

If I understood correctly both GMRES and GCR minimise the residual norm on $x_0+K_k$. Hence, they produce the same approximations over the iterations. The difference comes from the fact that GMRES ...

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