Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Infinite set A has the same cardinality as $\cup_{a \in A} B_a$

Hi all I am having difficulties in proving the following statement: Suppose $A$ is an infinite set, then there exists a bijection (for me, an injection would good enough) from $\cup_{a\in A} B_a$ to ...
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3answers
75 views

Check diagonalizability of a matrix without using eigen properties

For the matrix $$ A=\begin{bmatrix}0 & 1 & 0\\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} $$ How can we determine if $A$ is diagonalizable over $(a) \mathbb{F}^2 (b) \mathbb{Q} ...
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1answer
53 views

Common eigenvectors implies commutativity

I am stuck on a seemingly simple problem: if $\mathbf{M},\mathbf{N}$ are $n\times n$ and have all eigenvectors in common, then $\mathbf{MN}=\mathbf{NM}$. I can prove this if they are diagonalisable, ...
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51 views

how to write formal proofs involving nxn matrices

i have problems like these: Prove that if A is a nxn matrix, then tr(A-A^T)=0 Prove that if A and B are nxn matrices then tr(A+B) = tr(A) + tr(B). I can clearly understand why these hold true and ...
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33 views

Nullspace and Base of 2 by 2 Matrix [closed]

Hope someone can help me to solve and understand the following problem : $ f:M_{22} (\mathbb{R}) \rightarrow M_{22} (\mathbb{R}) $ $$A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ $$f(A) ...
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32 views

Column space of a matrix product

Let $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times q}$, $C\in\mathbb{R}^{n\times q}$ be matrices such that $$C=AB$$ where $C$ and $B$ are full column rank. Then do $C$ and $B$ have the same ...
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61 views

Why is this a linear transformation?

In class the professor said that f(x)-->g(x)*f(x) is considered a linear transformation. I don't understand why that is, can someone explain?
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50 views

Determinant question $\det(A^{-1/2}) = \det(A)^{-1/2}$

Can someone show me how: $\det(A^{-1/2}) = \det(A)^{-1/2}$ where we assume that $A$ is invertible. thanks
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2answers
70 views

A matrix with positive determinant

Let $A$ an $n \times n$ matrix with complex entries and $A^{*}$ its conjugate transpose. How can you show that the following $2n \times 2n$ matrix $\mathbf{X}=\left[\begin{array}{*{20}{c}} ...
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1answer
35 views

Linear operators and change of basis in a vector space

Suppose we have a vector space $V$ over a scalar field $\mathbb{F}$ and two different bases $\mathcal{B}=\lbrace\mathbf{v}_{i}\rbrace_{i=1,\ldots , n}$ and ...
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25 views

Calculate rotation on a sphere with given coordinates

I have a sphere with a fixed radius. I have a set of points on that sphere, let's say $p_1, p_2$ and $p_3$ and it's $3$D Cartesian coordinates. I rotated each of the points around the center of the ...
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62 views

Find the $L^2[-\pi,\pi]$ projection of $f(x)$

I need to find the $L^2[-\pi,\pi]$ projection of $f(x)=x^2$ onto the space $V_n\subset L^2[-\pi,\pi]$ spanned by ...
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0answers
35 views

Complex structure with a linear transformation

A complex structure on a real vector space $V$ is a linear endomorphism $J$ of $V$ such that $J^2=−1$, where $1$ is the identity transformation of $V.$ Let $T: V \rightarrow V$ be a linear ...
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2answers
44 views

Calculus on Matrices [closed]

I have a basic doubt regarding calculus involving matrices. Dimensions of each matrices are also indicated along matrix name Question If I have a matrix $\kappa(s)_{3\times 1}$ what is ...
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0answers
24 views

The space of alternating multilinear forms

I was just wondering if there is a standard (or even just usual) notation for the space of alternating $k$-linear forms on an $F$-vector space. I know that this space is naturally isomorphic to the ...
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0answers
26 views

Solving an equation with boundary conditions to find coefficients

I want to find the unknown constants in the function $f(x,y)=A(e^{-i.k_{x}x}+C_{1}x+C_{2})(e^{-i.k_{y}y}+C_{3}y+C_{4})$, using the following known boundary conditions and auxiliary equation ...
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1answer
34 views

Finding basis for polynomials with coefficients summing to zero

Let $\mathcal{P}_n[x]$ be the space of polynomials of degree $\leq n$ with real coefficients. I want to find a basis for the subspace of $\mathcal{P}_n[x]$ where the coefficents sum to zero, that is, ...
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1answer
27 views

Finding Euclidean Normal Form of an Isometry

Let $$A = \frac{1}{4} \left(\begin{matrix} \sqrt 3 + 2 & \sqrt 3 - 2 & -\sqrt 2 \\ \sqrt 3 - 2 & \sqrt 3 + 2 & - \sqrt 2 \\ \sqrt 2 & \sqrt 2 & 2\sqrt 3 \end{matrix} \right) ...
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4answers
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Proof by Induction - Algebra Problem (Steps included but not understood)

I do not quite understand this proof, if anyone could explain the steps for me it would be greatly appreciated. It's probably something glaringly obvious I'm not seeing, thanks in advance. Prove that ...
4
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1answer
189 views

Generate arbitrary numerically invertable matrix

I'm designing a unit-test for a matrix inversion function. Currently I make a random matrix as a test case by generating its elements with random numbers uniformly distributed in $[0,1)$. If I ...
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1answer
42 views

Equality in Cauchy-Schwarz Inequality implies linear dependence.

I need some help to show that: If $| \langle u,v \rangle | = \|u\| \|v\|$ then $u=\lambda v$ for some scalar $\lambda$. We have to consider this over an arbitrary field $\mathbb{F}$. I appreciate ...
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4answers
29 views

Let $X,Y,Z$ be subspaces of $V$ so that $X$ is a subspace of $Y$. Prove that $Y\cap (X+Z)=X+(Y\cap Z)$

Let $X,Y,Z$ be subspaces of $V$ so that $X$ is a subspace of $Y$. Prove that $Y\cap (X+Z)=X+(Y\cap Z)$ I know that I need to prove that $Y\cap (X+Z)\subseteq X+(Y\cap Z)$ and $X+(Y\cap Z)\subseteq ...
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Prove that $V = \ker(\phi) \oplus \text{image}(\phi)$

Let $V$ be a $n$-dimensional complex vector space and $\phi:V\to V$ a linear mapping. Prove that $$V = \ker(\phi^n) \oplus \text{image}(\phi^n)$$ Here is my attempt: Since $\phi^n$ is also a linear ...
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Tensor product of algebras which is Frobenius.

Let $A$ and $B$ be two finite dimensional algebras over a field $k$. Let us suppose that the $k$-algebra $A\otimes_{k} B$ is Frobenius (or symmetric). Is it true that $A$ and $B$ are two Frobenius ...
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Suppose a linear system of 4 equations in 4 unknowns has two distinct solutions. Prove that the system must have an infinite number of solutions.

Suppose a linear system of 4 equations in 4 unknowns has two distinct solutions. Prove that the system must have an infinite number of solutions. i've always been taught that if it has 2 solutions.. ...
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2answers
45 views

find the equation of the circle $x^2 +y^2 +ax +by = c$ passing through points $(6,8), (8,4), (3,9)$

Find the equation of the circle $x^2 +y^2 +ax +by = c$ passing through points $(6,8), (8,4), (3,9)$. How do I go about solving this? I don't have a textbook assigned so I'm not even sure what ...
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1answer
35 views

The bound of the largest eigenvalue of a symmetric positive definite matrix divided by its diagonal matix?

Suppose $A$ is a symmetric positive definite matrix, $D$ is the diagnal matrix of A. The largest eigenvalue of $D^{-1}A $ is denoted by $\lambda$. Then what is the bouned of $\lambda$? I only see ...
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1answer
38 views

Graph pruning whilst ensuring connectivity

Problem: I have a graph (in this instance, it's represented by a matrix which is $\in \mathbb{R}^{n \times n}$). In the raw graph, all nodes are connected to every other node (except themselves) in ...
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1answer
39 views

Convex matrix function

Please give me some hints for the following problem: Let $S = \{D \in R^{m \times n}, \|d_i\| \leq 1, i = 1, 2, \dots, n \}$. Find condition of $F \in R^{m \times m}$ such that the function: $ f(D) ...
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Showing $T: X\rightarrow Y$ is a linear map, is one-to-one… Over-thinking question?

so my question is as follows: Suppose that $X$ and $Y$ are normed linear spaces and that $T: X\rightarrow Y$ is a linear map (ie $T(\alpha x_1+\beta x_2) = \alpha T(x_1) + \beta T(x_2) \forall ...
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3answers
46 views

Let $(a,b), (c,d)$ be two vectors in $\mathbb R^2$. If $ad-bc=0$ prove that they are linearly dependent

Let $(a,b), (c,d)$ be two vectors in $\mathbb R^2$. If $ad-bc=0$ prove that they are linearly dependent My attempt: I tried to do it by contradiction: Suppose that they are linearly independent that ...
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1answer
73 views

Orthogonal complement of vector spaces

Let $V$ be a vector space. Here I do not restrict $V$ to be finite dimensional. Let $S$ be a vector subspace of $V$. Why is $S\subset (S^{\perp})^{\perp}$ rather than $S= (S^{\perp})^{\perp}$?
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Difference of Positive Semidefinite Matrices

Suppose I have two matrices: $$ A\succeq 0\\ B\succ 0 $$ and I know that $$ \langle v_i,Bv_i\rangle - \lambda_i \geq 0 $$ for every normalized eigenpair $(v_i,\lambda_i)$ of $A$. Is this enough to ...
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Orthonormal basis in $L^2$

Find an orthonormal basis in $L^2(-1,1)$ for $span\{1,x,x^2,x^3\}$. I know I must use Gram-Schmidt process in order to solve this problem. The answer given for the first vector is ...
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1answer
17 views

Distributing out log equation

$$\log_{27}x = 1 - \log_{27}(x-0.4)$$ $$\log_{27}(x(x-0.4))=1$$ $$x=5.4,\, x=-5$$ I'm confused on the second line. How come it is not $\log_{27}(x+x-0.4)$?
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Prove that $\dim(U_{\perp}) = \dim(V ) − \dim(U)$.

Let $V$ be a finite-dimensional inner product space over field $F$, and let U be a subspace of $V$ . Prove that the orthogonal complement $U_{\perp}$ of $U$ with respect to the inner product $\langle ...
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1answer
176 views

Cube of an integer

$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=k$ and $x, y, z, k$ are integers. Prove that $xyz$ is cube of some integer number. I was wondering about giving a parametrization for the rational points on ...
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2answers
28 views

Norms and orthogonality

I am trying to refresh my memory with linear algebra. I am confused with these set of questions: Show that $a=\begin{pmatrix} 1 \\ i \\ -1 \\ -i \end{pmatrix},$ $b=\begin{pmatrix} ...
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1answer
30 views

Orthogonal complement is the set of vectors orthogonal to the rows of matrix A. Why?

When reading about orthogonal complement, I encounter the following claim: If the subspace is described as the range of a matrix: $S = \{ Ax : x \in \mathbf{R}^n \}$, then the orthogonal ...
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1answer
21 views

Question on a set of transpose matrices and linear independence

Question: Prove that if {$A_1,A_2,\cdots,A_k$} is a linearly independent subset of $M_{n\times n} (F)$, then {$A^t_1, A^t_2,\cdots, A^t_k$} is also linearly independent. My attempt: I know that the ...
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1answer
73 views

Intuitive meaning of the exponential form of an unitary operator

I'm an undergraduate student in Chemistry currently studying quantum mechanics and I have a problem with unitary transformations. Here in my book, it is stated that Every unitary operator ...
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25 views

rank of block matrics

Let $H=\pmatrix{A_{i\times i}& B_{i\times n-i} \\\ C_{n-i\times i}& D_{n-i\times n-i}}\in M_n(\mathbb F)$ where $\mathbb F$ is a field. Suppose that $rank(H)\geq n-i$ and $XB+YD=I_{n-i\times ...
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0answers
43 views

Matrix rotations

I am trying to find an angle to rotate the basis $$\left[ {\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} } \right]$$ to $$\left[ {\begin{array}{cc} -1 & 1 \\ 1 & 1 \\ ...
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4answers
36 views

Inverting a vector

If I have $Ax=b$ where $A$ is $n$ by $n$ while $x$ and $b$ are $n$ by 1, is it possible to find $A$ given $x$ and $b$. The idea would be some sort of $x^{-1}$ operation on the right of both equations ...
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2answers
55 views

Eigenvalues of a special $M \times M$ matrix

I could not obtain an explicit formula for the eigenvalues of matrix $$ \begin{pmatrix} a & b & 0 & 0 & 0 & \cdots & 0 \\ c & a & b & 0 & 0 & \cdots & 0 ...
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At what time and distance from Delhi will the mall train completely cross the goods train?

A goods train $158$ metres long, and traveling at the average speed of $32$ km/hr leaves Delhi at $6:00$ A.M. Another mall train $130$ metres long and traveling at the average speed of $80$ km/hr ...
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2answers
36 views

What does it mean when dim(V)=rankT

I have a question relating to a linear transformation and have ended up with the result that $dim(V)=rank(T)$. I got to this because I'm told that $V$ and $W$ are finite dimensional vector spaces, ...
2
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1answer
23 views

The computation of nullity and rank of a linear transformation.

Let $V$ be the linear space of all real functions continuous on $[a, b]$. If $f\in V, g=T(f)$ means that $$g(x)=\int_a^b f(t)\sin(x-t)\,dt\hspace{1 cm} for\ a\le x\le b$$ We can see this as a linear ...
2
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0answers
29 views

Constrained Quadratic Optimization(Reproducing Kernel)

I am attempting to use a constrained quadratic optimization to find the coefficients of a reproducing kernel. The problem is as follows: $y(t)=\sum_{i=0}^J\alpha_iK(t, t_i)$ $Q(\alpha)= ...
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3answers
30 views

Prove that $\exists y\in V$ so that the set {${u+y:u\in U}$} is a subspace of $V$

Let $V$ a vector space over a field $F$, and let $v,w\in V$ so that $v\neq w$. Define $U=${${(1-t)v+tw: t\in F}$}. Prove that $\exists y\in V$ so that the set {${u+y:u\in U}$} is a subspace of $V$ ...