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

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real numbers a vector space over rational numbers?

Let $V$ be set of real numbers and $K$ the field of rational numbers. Is $V$ a vector space over $K$, with ordinary addition of real numbers and multiplication by rational numbers?
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About $\mathcal{L}(V,W)$

Let $V,W$ are two vector space and let $S\subseteq V$. Define: $$S^{0}=\{T\in\mathcal{L}(V,W)\mid~T(x)=0, \forall x\in S\}$$ The problem aks me to verify $S^{0}$ is a subspace of $V$ and if $V_1,V_2$ ...
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14 views

Invariant hermitian forms and irreducible representations

Let $V$ be a vector space over $\mathbb{C}$ of finite dimension $n$, $G$ is a finite group and $T:G\rightarrow GL(V)$ its irreducible representation that sends each $g$ into $T_g$. Let $E:V^{\bigoplus ...
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25 views

Minimum polynomial and matrix multiplication

May you help me with the following proving? Let $A,B$ be square matrices over $\mathbb C$ and suppose that there exist rectangular matrices $P,Q$ over $\mathbb C$ such that $A=PQ$ and $B=QP$. ...
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33 views

Minimal polynomial of a linear operator

How to show the following: Let $T: V_F \to V_F$ be a linear operator and $f(x)$ be the minimal polynomial of $T$ over $F$. Let $$f(x)=g_1(x)g_2(x)\cdots g_n(x)$$ where the $g_i$'s are monic and ...
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23 views

Composition of systems of equations

Suppose $$2x + 3y = u$$ $$x - 4y = v$$ and further that $$3u - 5v = c$$ $$2u + 3v = d$$ Express c and d in terms of $x$ and $y$ by matrix multiplication. It's quite easy by direct substitution but ...
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Using a matrix to organise values into groups

Let's say I have a matrix of size 6 x 6. Six students are 'ranking' six other students (including themselves). If I wanted to organise them into let's say, groups of three without picking and ...
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Symmetric Matrices of $I_{2}$

Find 10 symmetric matrices $ A = \left| \begin{array}{cc} a & b \\ c & d \\ \end{array} \right|$ such that $A^{2}=I_{2}$ (I'm going to call matrix A the "square root" of $A^{2}$. If this is ...
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Matrix multiplication related to complex numbers?

Evaluate and simplify the product $\begin{bmatrix} r\cos(\alpha) & -r\sin(\alpha) \\ r\sin(\alpha) & r\cos(\alpha)\\ \end{bmatrix}$ $\begin{bmatrix} s\cos(\beta) & -s\sin(\beta) \\ ...
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Having trouble using eigenvectors to solve differential equations

The question asked to solve $$\frac{dx}{dy} = \begin{pmatrix} 5 & 4 \\ -1 & 1\\ \end{pmatrix}x$$ ,where $$ x = \begin{pmatrix} x_1 \\ x_2 \\ \end{pmatrix}$$ I went ...
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37 views

Does convex and radially open imply open?

I want to show that a convex set $A$ is radially open iff $A\cap W$ is open in W, for every finite dimensional linear subspace. Here the 'openness' we are talking about is from any normed space. ...
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How to show that a valid inner product on V is defined with the formula $[x, y] = \langle Ax, Ay\rangle $?

Let $A \in L(V,W)$ be an injection and $W$ an inner product space with the inner product $\langle \cdot,\cdot\rangle $. Prove that a valid inner product on $V$ is defined with the formula $[x, y] = ...
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46 views

Groups of transformations

I tried to find literature and articles about groups of transformations, but mostly what I found is either about groups or about transformations. Can you suggest me literature where groups of ...
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1answer
67 views

Given a vector space $V$, show that the following statements are equivalent.

Given a subset $W$ of $V$ then I want show that, $\forall v \in V, w \in W$ $\exists \lambda \in \mathbb{R}$ such that $w + \alpha v \in W$ for any $0 < \alpha < \lambda$ iff $\forall v \in ...
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1answer
24 views

how would $f(T)$ look like if …

I know the result that if $T:V_F\to V_F$ is a linear operator then for any polynomial $f(x)\in F[x],~f(T)$ is a linear operator. Now my question is how would $f(T)$ look like if $f(x)$ is the zero ...
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Fast way to calculate Eigen of 2x2 matrix using a formula

I found this site: http://www.math.harvard.edu/archive/21b_fall_04/exhibits/2dmatrices/index.html Which shows a very fast and simple way to get Eigen vectors for a 2x2 matrix. While harvard is quite ...
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33 views

calculate kernels of matrices with angles

So my professor gave me this question: I have to find the basis of the eigenvalues of this matrix \begin{pmatrix} \cos(q) & \sin(q)\\ \sin(q) & -\cos(q)\\ \end{pmatrix} so I calculate ...
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27 views

Consequences of a rectangular matrix being of maximal rank

I have a real matrix $A$, $(m+1) \times m$ and a vector $b \in \mathbb R^{m+1}$ such that $b_{m+1}=0$. For any vector $u\in \mathbb R^m$, $Au=0 \Rightarrow u=0$. This means that $A$ is a rectangular ...
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23 views

Some remarks/questions from Primary Decomposition Theorem to get verified

In course of self-studying the Canonical Form in Linear Algebra I'm trying to put some remarks from the concept I acquired from Primary Decomposition Theorem which reads as follows: Let $T:V_F\to ...
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46 views

Cases where characteristic and minimum polynomial coincide

Given a matrix such as $$\pmatrix{0 & 0 & 2 \\ 1 & 0 & -1 \\0 & 1 & 1 \\ },$$ whose characteristic polynomial is $-X^3+X^2-X+2.$ $$$$How it could be deduced that it equals ...
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48 views

Vectors that form a triangle!

I have a problem here. How can I prove that sum of vectors that form a triangle is equal to 0 (AB+BC+CA=0) ? Thank you!
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1answer
45 views

Is the inverse function smooth?

Imagine that we have a function $Inv$ that maps $A \rightarrow A^{-1}$, where A is an invertible square matrix. now my questions is: how do i see that this function is arbitrarily often ...
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34 views

Prove if we have a square unitary Matrix $Q$, then $\det(Q) = e^{i\theta}$

Prove if we have a square unitary Matrix $Q$, then $\det(Q) = e^{i\theta}$ Using $\det(Q)\det(\bar{Q}^T) = I$, I get to the stage $\det(\bar{Q})\det(Q)=1$, but can't do much else with it. Thanks for ...
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Characteristic polynomial of the differentiation map

Determine the characteristic and minimum polynomial of the differentiation map $D: \mathbb{R_n}[X]\longrightarrow \mathbb{R_n}[X]$ (where $\mathbb{R_n}[X]$ is a set of polynomials of degree at most ...
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3answers
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How to form a cubic equation with the substitution method?

I had this question: "Find the cubic equation whose roots are twice the roots of the equation $3x^3 - 2x^2 + 1 = 0$" In my first attempt, I solved it through the use of simultaneous equations, where ...
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1answer
38 views

Arrangements of affine hyperplanes

Fix $n>0$ and $X\subseteq\mathbb{R}^n$. Call a function $f:X\longrightarrow \mathbb{R}$ linear if it is of the form $$ f(\bar{x})=a_1x_1+\ldots+a_nx_n+b $$ for some $a_i,b\in\mathbb{R}$. Now ...
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36 views

Dimensions of vector subspaces in a direct sum are additive

$V = U_1\oplus U_2~\oplus~...~ \oplus~ U_n~(\dim V < ∞)$ $\implies \dim V = \dim U_1 + \dim U_2 + ... + \dim U_n.$ [Using the result if $B_i$ is a basis of $U_i$ then $\cup_{i=1}^n B_i$ is a basis ...
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49 views

Is it possible that $V=\cup_{i=1}^n V_i?$

Let the vector space $V$ be decomposable into the non-zero subspaces $V_i;~i=1(1)n.$ Is it possible that $V=\cup_{i=1}^n V_i?$
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Extending a rational entry matrix to an orthogonal matrix.

Suppose $M \leq N$ are positive integers and let $r_1,\ldots,r_M$ represent the rows of an $M\times N$ matrix $A$ in which: (i) the rows of $A$ are orthogonal and having the same norm, (ii) the ...
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25 views

Matrix involving values of polynomials

I've been doing this problem but im stuck. Be $f_1 f_2 f_3 \in \mathbb{R}_2$[$x $]. Proove that {$f_1$,$ f_2$,$ f_3$} form a base of $\mathbb{R}_2$[$x $] as $\mathbb{R}$ vector space, if and only ...
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1answer
41 views

A question on linear operators

This is a problem I’ve been working on as part of my studies for an upcoming comprehensive exam: Let $F$ be a field, let $V\in F$-$\mathrm{Mod}$ be a finite-dimensional left $F$-vector space, and let ...
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27 views

Linear Algebra determinant and rank relation

True or False? If the determinant of a $4 \times 4$ matrix $A$ is $4$ then its rank must be $4$. Is it false or true? My guess is true, because the matrix $A$ is invertible. But there is ...
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2answers
32 views

Linear Transformation Orthogonality

True or False: If $T$ is a linear transformation from $R^n$ to $R^n$ such that $$T\left(\vec{e_1}\right), T\left(\vec{e_2}\right), \ldots, T\left(\vec{e_n}\right) $$ are ...
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1answer
48 views

What are the two main ways to prove that a matrix $\Bbb R^{n\times n}$ is definite positive.

What are the two main ways to prove that a real $n\times n$ matrix is definite positive? Is the first way: If a matrix is $n \times n$ symmetric matrix, then the associated quadratic form is ...
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1answer
48 views

Quadratic Equation with “0” coefficients

Let's say I have two objects $x$ and $y$ whose position at time $t$ is given by: $$ x = a_xt^2+b_xt+c_x \\ y= a_yt^2+b_yt+c_y $$ And I want to find which (if any) values of $t$ cause $x$ to equal ...
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1answer
20 views

Finite difference method stability

I have shown that a finite difference method satisfies $$\underline{u}^{n+1}=((1+6\mu)\mathbb{I}-36\mu A^{-1})\underline{u}^n$$ I don't think that the rest of the question is necessary but it is ...
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1answer
27 views

The relationship between plane curves and the derivative of the Wronskian

I have found a theorem but I did not understand the proof. I'm looking for a clarification of the proof or a different proof. Let $f_1, f_2, f_3$ be the three components of a curve in $R^3$ ...
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61 views

if $\mathbf x$ is sampled randomly from a hypercube on $R^n$, what is the probability density for $|\mathbf x| = d$

if the vector $\mathbf x$ is sampled randomly from a uniform distribution on $[0, 1]^d$, what is the probability density function for $|\mathbf x|$? Is it easy to scale for $[0, n]^d$?
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2answers
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Showing set of tensored states span a space

I have the four states $$ \lvert1\rangle \lvert1\rangle - \lvert0\rangle \lvert0\rangle \\ i\lvert1\rangle \lvert1\rangle + i\lvert0\rangle\lvert0\rangle \\ \lvert0\rangle \lvert1\rangle + ...
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2answers
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Smooth maps on a manifold lie group

$$ \operatorname{GL}_n(\mathbb R) = \{ A \in M_{n\times n} | \det A \ne 0 \} \\ \begin{align} &n = 1, \operatorname{GL}_n(\mathbb R) = \mathbb R - \{0\} \\ &n = 2, \operatorname{GL}_n(\mathbb ...
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Conditions for matrix operator to preserve complex symmetry on DFT vector?

Suppose there is a DFT vector $\mathbf{X}$ (complex vector) with length N, which presents complex conjugate symmetry around its middle point, i.e., $X(1) = X(N-1)^*$, $X(2) = X(N - 2)^*$ and so forth. ...
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Why don't we consider the zero subspace (which is readily $T$-invariant) in the definition of direct sum of linear operators?

Why don't we consider the zero subspace (which is readily $T$-invariant) in the definition of direct sum of linear operators? REF: Schaum's Outline of Linear Algebra
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Book on quadric surfaces with linear algebra

Most information that I can find about quadric surfaces is written from a calculus perspective - without using any matrices or vectors. However, I would like to have a reference that tells me the ...
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Eigenbasis of a Hilbert space: isomorphism

Let $K$ be a matrix containing the dot product between points in a Hilbert space $\mathcal{H}$ (assume that it is finite-dimensional). Then, we could form a basis using the eigenvectors of a normal ...
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Solving the domain and range of a region satisfying two inequalities?

The question I was provided was: "Find the domain and range of the region satisfied by the following inequalities: i) $y \ge (x-1)^2$ ii)$y \le2x+1$ Any help would be greatly appreciated. Would you ...
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please help me with this question [duplicate]

How can I solve that? Thanks! Let V = span{1, x, x2 , x3 } be a real inner product space with the inner product defined by (f, g) = integral from -1 to 1 (fg)dx. Check that φ(f ) = f (0) is a linear ...
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72 views

inner product space and polynomial

Let $V = \mathrm{span}\{1,x,x^2,x^3\}$ be a real inner product space with the inner product defined by $$ \langle f,g\rangle =\int\limits_{-1}^{1} fg $$ Check that $T(f) = f(0)$ is a linear ...
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1answer
24 views

Column entries of a matrix sum to zero, so what are the properties?

What kind of properties does a matrix whose column entries sum to zero have? $$ \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & ...
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29 views

Resolving mass of holy disk with moment of inertia?

A uniform lamina of mass m is bounded by concentric circles with centre O and radii a and 2a. the lamina is free to rotate about a fixed smooth horizontal axis T which is tangential to the outer ...
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99 views

Linear equations; real solution; rational solution?

I saw this question Let $A ∈ M_{m\times n}(\mathbb{Q})$ and $B ∈ \mathbb{Q}^m$. Suppose that the system of linear equations $AX = B$ has a solution in $\mathbb{R}^n$. Does it necessarily have ...

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