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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Find the given binary matrix.

Consider a binary square matrix $A$ ,(that is elements are either $1$ or $0$). You are given $Sum$ of elements of $each$ $row$ and $column$ ,that is you are given $r_i$ where $r_i$ means $Sum$ of its ...
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Orthogonality and Linear Independence | Intuition

I just like to conceptually understand linear independence and orthogonality. So in reality do one tests for whether say two vectors are co-linear and the other tests whether two vectors are ...
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$\forall n \geq 3, \text{Span}(SO_n(\mathbb{R})) = M_n(\mathbb{R})$

I would like to prove that the following holds for all $n \geq 3$ : $$\forall n \geq 3, \text{Span}(SO_n(\mathbb{R})) = M_n(\mathbb{R})$$ So far I have noticed the following : For $n =2$ the ...
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Adding an element in a vector space with an element from a proper subspace of that vector space

Let $S$ be a proper subspace of a vector space $V$ and let $\vec{x} \in S$ and $\vec{y} \in V, \notin S$. Clearly $\vec{x} + \vec{y} \in V$, by closure of $V$ under addition, but can we have $\vec{x}...
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$E\cong E^{**}$ then $\dim E<\infty$

This is an exercise from Trèves book, Topological Vector Spaces, Distributions and Kernels. 2.8. Prove that every vector space $E$ is isomorphic to the direct sum of a family of one-dimensional ...
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9 views

Spectral norm of random matrices times a diagonal matrix

Consider some scalars $d_1\leq \dots\leq d_k$ and let $D = \text{diag}(d_i)\in\mathbb{R}^{k\times k}$. Assume that $X\in \mathbb{R}^{k \times k}$ is a random matrix of i.i.d. normal entries (i.e. $X_{...
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20 views

Existence of a continuous linear functional with $f(x_0)=||x_0||$ and $ \sup\{|f(x)|:x\in X,x\neq 0\}=1$

Let X be a linear, real or complex normed space and let $x_0\in X,x_0\neq0$. Show that: It exists a continous linear functional $f:X\rightarrow\mathbb{K}$ where $\mathbb{K}=\mathbb R,\mathbb C$ s.t :$...
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54 views

$\forall \lambda \in \mathcal{F}: A x=\lambda x: \lambda \in \mathbb{R}$ - Correct notation?

I am new to formal mathematics, and thus not very experienced with correct notation. Consider $A\in End(V)$ where $V$ is a vector space over $\mathcal{F}$. I want to write $\sigma(A)\subseteq \mathbb{...
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Homogeneous linear differential equation with constant coefficients.

Is the following statement true? Any finite-dimensional subspace of $C^\infty=\{f\in \mathcal F(\Bbb R,\Bbb C) \;|\; f \text{ has derivative of all orders.}\}$ is the solution space of a ...
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1answer
27 views

Eigenvalues of a block matrix with all diagonal blocks but one

Let us consider a block matrix of the form $$A= \begin{bmatrix} -(k+\mu)I & B \\ kI & -(\gamma + \mu)I \end{bmatrix},$$ where $I$ is the $n\times n$ identity matrix, $\gamma, k$ and $\mu$ ...
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Prove $(\forall x, y \in E)(\exists ! z\in Z)$ s. t. $x + z = y$.

I am currently studying Introduction to Hilbert Spaces with Applications, the third edition, by Debnath and Mikusinski. Chapter 1, exercise 1, is as follows: Prove that for every $x, y \in E$ ...
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Finding the Quadratic Form of the Difference of Eigenvectors

So I got this problem for homework and was able to finish part a where I need to draw an eigenvector on a level curve but I can't solve part b I'm assuming I need to separate it into $qA(v_1)-qA(v_2)$...
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1answer
16 views

Computing transformation matrix of line segment

Consider a line segment $L_1$ defined by two points $A_1$ and $A_2$ in $3D$ space and another line segment $L_2$ defined by $B_1$ and $B_2$ where the length of these line segments are identical. What ...
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How Do I Find Eigenvectors and Eigenvalues Reflected About a Plane?

Trying to solve this problem, but I don't even know where to start? Can I get some help? Here's the problem: Let $A$ be the matrix of the linear function $T: \mathbb R_3 → \mathbb R_3$ that reflects ...
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25 views

Calculate the orthogonal projection onto a vector $v$

I have to calculate the orthogonal projection of the vector $$ v = \begin{pmatrix} 2 \\ 4 \\ 2 \end{pmatrix} $$ on the $\text{span}(u_1,u_2)$ where $$ u_1 = \begin{pmatrix} 1/2 \\ 0 \\ 1/2 \end{...
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15 views

Pseudoinverse of a matrix with columns of exponential decays

I want to calculate the pseudoinverse $A^+$ of a matrix $A$ whose columns are exponential decays: $$ \begin{pmatrix} e^{-\alpha_{0}t_{0}} & e^{-\alpha_{1}t_{0}} & e^{-\alpha_{2}t_{0}} ...
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1answer
22 views

Determinant and norm of the cross product

I assume the vectors are in Euclidean space. I know that the determinant of a vector family is the area/volume of the associated parallelogram/parallelepiped. But I also read that the norm of the ...
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2answers
26 views

Short exact sequence of vector spaces

Suppose $V$ is a finite-dimensional vector space, and $U,W$ its subspaces. Is it possible to construct the following short exact sequence? $$0\rightarrow U\cap W\xrightarrow{f}U\oplus W\xrightarrow{g}...
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Stationary distribution in a Markov process.

Consider the homogeneous Markov process with matrix $W$ which describes the "probability of transition" to pass from a state $a$ to $b$. So in the time $t+1$ the probability to be in the state $a$, $...
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What is this tensor operation called?

I'm wondering if the following operation has a name: $$ \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{matrix} ? M = \begin{matrix} aM ...
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41 views

Does there exist a non-symmetric involutory matrix?

Let $A$ be a real involutory matrix i.e. $$A^2 = I.$$ Is it necessarily symmetric? Any help will be highly appreciated. Thank you very much.
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Show that there is only one linear function that satisfies $f(b) = g(b)$

Let B be a basis of a vector space $V$ over a field $\mathbb{K}$ and $W$ another vector space over $\mathbb{K}$. Let $g: B \to W$. Show that there is exactly one linear function $f: V \to W$, such ...
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1answer
18 views

What is the probability that a uniformly random vector is spanned by a subset of basis vectors?

Working in the set of Real Numbers. Say that you're given a set of basis vectors $\{\mathbf{b}_i\}, \, i \in [n]$. Now, sample a vector v uniformly at random from the entire space. What is the ...
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15 views

Bounding the norm of the inverse matrix

Very strangely, I do struggle with answering the following, seemingly elementary, question. For a square matrix $A$, is there an upper bound for the norm of its inverse? In terms of e.g. the norm of ...
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36 views

How can we find length of v in inner space product using basis

Consider R³ as inner product space with standard inner product. Let W be the subspace of R³ with basis B = {(1,0,−2),(−3,2,1)} and let v = (−1,2,−3) be in W. (i) Find the length of v directly. (ii) ...
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43 views

Are orthogonal operators always isomorphisms?

I need to show the following: Let $V$ be a finite dimensional vector space with inner product, if $T$ is orthogonal, show that T is injective and surjective I think it is injective because T ...
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16 views

Commutator of 2 Rotational Matrix in x and z axis

How do i find the commutator of rotational matrix in x and z axis? Dx = \begin{pmatrix}1&0&0\\ 0&cos\gamma &-sin\gamma \\ 0&sin\gamma &cos\gamma \end{pmatrix} Dz = \begin{...
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32 views

How to solve $vv''+v'^2-6t^2=0$

Hi, I have been trying to solve this equation for a week now. However, I keep ending up with the same result everytime, it may be because my knowledge on this chapter is not great. I am unable to ...
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57 views

Show $G(T^k,\lambda^k) \subseteq G(T,\lambda)$ for generalised eigenspaces.

Let $T$ be a linear operator on complex vector space $V$ and $G(T,\lambda)$ denote the $\lambda$ generalised eigenspace for $T$. Show $G(T^k,\lambda^k)= G(T,\lambda)$. I have trouble proving $G(T^k,...
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Find the best $X$ to minimize the diagonal entries of $A^TXA - D$, allowing arbitrarily large off-diagonals.

Given an (arbitrary) matrix $A \in R^{m \times n}$ and a diagonal matrix $D \in R^{n \times n}$, I want to find a matrix $X \in R^{m \times m}$ such that the diagonal entries of $A^TXA$ "best ...
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1answer
53 views

Determinant of $2\times 2$ block matrices

I am trying to solve the problem here: Let $A,B,C,D,$ be commuting $n\times n$ matrices over the field $F$. Show that the determinant of the $2n\times 2n$ matrix $$\begin{bmatrix} A&B\\C&D \...
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1answer
9 views

Proving postulate about a property fo spherical vectors

Assume we have $X, Y$ constant unit vectors of $\mathbb{R}^3$ I postulate that the maximum of the function: $(V \cdot X) (V \cdot Y)$ I reached by the halfway vector between $X,Y$ i.e at the vector ...
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26 views

How are stronger $H^n \subseteq P$ or $H \subseteq \sqrt{P}$?

Let $L$ be a Lie algebra over k and $H, P$ are ideals of Lie algebra $L$ Set $H^1:=H$. Define recursively $H^{n+1}:=[H,H^n]$ for all $n≥1$. Then $H^{n+1}=[H,H^n]⊆H^n$ for all $n$. $\sqrt{P}= \cap \{...
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Linear image of almost disjoint union is almost disjoint?

Suppose I have some boxes $B_1, ..., B_k$ in $\mathbb{R}^n$. (By box I mean a product of closed intervals.) We say that the union $\bigcup_{i=1}^k B_i$ is almost disjoint if the interiors of the boxes ...
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Lanczos Iteration recurrence

While stuyding Lanczos Iteration in "Matrix Computations", I don't understand the recurrence relation in Theorem 10.1.1. $$AQ_k = Q_kT_k + r_ke_k^T \quad for\quad k=1:m \tag{1}$$ The author derives ...
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Relation between Cayley Transform and symmetric operators

I have been thinking about the following problem from section 10.6 of Kreyszig's book: If the Cayley transform $U$ of a symmetric linear operator $T:D(T)\to H$, where $H$ is a Hilbert space, is ...
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42 views

Is it true that $T$ is orthogonal if and only if $T$ is isomorphism?

I want to prove the following: Let $V$ be a finite dimensional vector space with inner product, then $T$ is orthogonal if and only $T$ is an isomorphism I think the sufficiency could be true because ...
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1answer
46 views

If T ◦ S = S ◦ T and v is an eigenvector of T, then v is an eigenvector of S

I have to prove or provide a counterexample to the following claim: Let $V$ be a vector space, $T, S : V \to V$ linear transformations. If $T S = ST$ and $v$ is an eigenvector of $T$, then $v$ is an ...
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Proof that the difference of any two semi-simple matrices is also semi-simple

Let $A$ and $B$ be semi-simple matrices. For $A-B$ to make sense, $A$, $B$ and $A-B$ need to be matrices of the same size $n$ over the same field $F$. If $F$ is algebraically closed, like when $F=\...
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1answer
25 views

Prove\Disprove: $TS=ST$, and $\mathbf{v}$ is an eigenvector of $T$ corresponding to eigenvalue $\lambda$ Then $\mathbf{v}$ is an eigenvector of $S$.

Prove or Disprove: Let $V$ be a vector space of finite dimension, and $T,S:V\to V$ linear maps. Suppose $T\circ S=S\circ T$, and that $\mathbf{v}$ is an eigenvector of $T$ corresponding to the ...
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2answers
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What exactly is the Metric Tensor?

I have recently watched (part of) a video by DrPhysicsA on Einstein's Field Equations. Now, he starts right off by explaining the Metric Tensor is through lots of equations that made general sense. ...
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$A$ is a square matrix and $\ker A^{n-1} \neq \ker A^n$ where $n≥2$ [closed]

I struggle with answering the question in preparation for the linear math exam. In class we haven't touched F and worked mainly on R. $A$ is a square matrix and $\ker A^{n-1} \neq \ker A^n$ where $n≥...
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1answer
62 views

How to find the determinant A from an equation having A as variable?

I'm currently struggling because I can't find the answer do this. If anyone can help me, it would be great. A is a $5\times 5$ non scalar matrix, $(A+2)(A+A^3+1)^2 (A^2+A^3+1)^3 =0 $ a) ...
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Let $\sigma$ be PSD. Is it always possible to find a basis of operators $\{A_i\}_i$ such that $tr(A_i^\dagger A_j \sigma) = 0$ unless $i = j$?

If so, how would this be proved? (Assuming that the Hilbert space is finite-dimensional, if that's relevant.) I was thinking that if $\sigma$ were positive definite, then $\mathrm{tr}(A_i^\dagger ...
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1answer
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Why does $T(x)=2x$ keep angles but is not an isometry? [closed]

I would like to know why a defined operator $T$ on $V$, as following: $T(\alpha)=2\alpha$ preserves angles but is not an isometry?
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1answer
19 views

If original set of vectors have zero mean, will the orthogonally projections of the vectors onto another vector have zero mean?

Consider vectors $x_1, \cdots, x_n \in \mathbb{R}^m$. Define the vector $\mu \in \mathbb{R}^m$ to be the mean of the vectors: $$ \mu = \frac{1}{n}\sum_{i=1}^n x_i $$ Assume that $\mu = 0$, the zero ...
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56 views

find all pairs of natural numbers $(x, y)$ so that $121$ divides $x^2+y^2$

I have to find all pairs of natural numbers $(x, y)$ so that $121$ divides $$x^2+y^2$$ I thought of writing the amount as equal to $11z$, $z$ natural number. Then I noticed that the numbers must have ...
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16 views

Matrix Operations: Name of function to rotate each column such that the first element is on the main diagonal

Suppose I have a matrix, A = [ 1 1 1 2 2 2 3 3 3 ] And I would like to create or find some function g such ...
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18 views

Is $\mathfrak{c}E$ a $C$-module for an abelian ring $C$ and an ideal $\mathfrak{c}$?

Let $C$ be an abelian ring, $\mathfrak{c}$ an ideal of $C$ and $E$ a $C$-module. Let $\mathfrak{c}E$ be the sub-$\mathbf{Z}$-module of $E$ generated by the family $(c x)_{(c,x)\in\mathfrak{c}\times E}$...
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20 views

Jordan Canonical Form test for diagonalization

I'm trying to prove that if we have $T$ linear operator with finite dimension, then the canonical jordan form of $T$ is a diagonal matrix if and only if $T$ is diagonalizable. I can see that it's ...

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