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 ...

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Maximizing inner product of unit vectors

I am working on a research project and the main algorithm is based on computing the following function: Given a symmetric matrix $A$ and a unit vector $x_0$, compute: $\max\langle Ax,x\rangle : ...
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Prove there is a unique $x_0 \in M$ such that $\|x_0 - z\| = \inf_{x \in M} \|x-z\|$ and show $z-x_0 \perp M$

Prove there is a unique $x_0 \in M$ such that $$\|x_0 - z\| = \inf_{x \in M} \|x-z\|$$ and show $z-x_0 \perp M$. Here, $X$ is a finite dimensional inner product space and $M$ is a proper, nonempty ...
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Wedge product and its dual

I am learning about differential forms and exterior algebra, and I am trying to get more familiar with the wedge product of vectors. A differential form is an element of $\left( \bigwedge^k ...
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Solving a common tangent problem using matrices

Provided only with the radii and centers of two circles. Set up a system of equations for the points of tangency such that it can be solved by an iterative, numerical, method. I have found this ...
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Given several vectors, how do you find linear transformations with a kernel/image that spans those vectors?

Given several vectors, how do you find the matrix of a linear transformations with a kernel/image that spans those vectors? I'd like to know how to find the matrix of a linear transformation given ...
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16 views

associated matrix to an orthonormal basis

Let T be a symmetric bilinear form. Given an orthonormal basis for the vector space, is the associated matrix the identity matrix? Thanks in advance.
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16 views

Dependent and independent variables of a differential equation

$$\theta'' + \delta\theta'+\sin\theta = F\cos(\omega t)$$ I am trying to write it as a first order equation, and state the dependent, independent and parameters in the ODE. I have written it as: ...
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29 views

Verify if this set of matrices span $M_2(\mathbb{R})$

I have the set: $$S = \left\{\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}, \begin{bmatrix}1 & 1\\0 & 0\end{bmatrix}, \begin{bmatrix}0 & 0\\1 & 1\end{bmatrix}, \begin{bmatrix}0 ...
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8 views

Finding coordinate axis vectors to yield a specific transformation

For a project I am working on, I will be given a point $\textbf{a}$ in a global 3-D coordinate system, as well as a point $\textbf{a}^\prime$ in a local coordinate system. (The global coordinate ...
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20 views

A Basis for a Jordan Normal Form

In my assignment I have to find a Jordan normal form for this matrix: Thank you for your help, and I'm sorry the question is pronunced with Latex.
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24 views

Invertible Matrices and Linear independence

If a matrix is invertible, what does this tell us application wise? I am familiar with what it implies in regards to the properties of the matrix, i.e: the determinant is non-zero, and for a matrix ...
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Prove that if rank(A + B) = rank(A) + rank(B), then col(A) ∩ col(B) = {0}

Let A,B be in Mmxn(R) Prove that if rank(A + B) = rank(A) + rank(B), then col(A) ∩ col(B) = {0} I started with a proof by contradiction, since we know that rank(A ...
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36 views

Knowing the eigenvalues for A find the matrix A

I know the eigenvalues for a matrix. Let's say they are 2 and 1. How can I find the matrix A for them (all members of A are not null) ?
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1answer
24 views

How can I find values that make a matrix Linearly Dependent?

Let $$A=\left(\begin{smallmatrix}2&-1&0&1\\0&0&a&3\\0&0&0&b\end{smallmatrix}\right)$$ For what choices (if any) of real numbers a and b are the rows of A linearly ...
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Let $F$ and $G$ be linear transformations. Find the transformation matrix in respect to basis $B$ and $C$. (polynomials)

Let $F$ and $G$ be two linear transformations that maps from $P_2(\mathbb{R})$ to $P_3(\mathbb{R})$, such that: $$F(p(t)) = tp(t)-p(1)\\G(p(t)) = (t-1)p(t)$$ Find the transformation matrices of $F$ ...
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1answer
11 views

Infinity norm greater $1$ implies that spectral radius greater $1$?

Suppose I have an arbitrary real matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ such that the sum of absolute values in each row is greater than $1$: $$\sum_{j=1}^n |A_{ij}|>1,\quad \forall ...
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3answers
61 views

Rank of $ A^2 +A + I$

Let $A$ be $6 \times 6$ real symmetric matrix of rank $5$. Find the rank of $A^2 +A +I$. Well i dont know any tool that can solve this question. The book says answer is $5$ but it could be wrong ...
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27 views

Showing that something is the zero matrix

I'm trying to prove that if the linear system of equations $Ax=b$, where $A$ is a matrix with $m$ rows and $n$ cloumns, only has a solution when $b=0$ then $A$ is the zero-matrix. My idea goes like ...
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25 views

How to prove $A( v \times w)$ = $\det A\cdot(Av \times Aw)$ , where $A\in \text {Mat}_{3 \times 3}(\mathbb R)$ is an orthogonal matrix.

How to prove $A( v \times w)$ = $\det A\cdot(Av \times Aw)$, where $A\in \text {Mat}_{3 \times 3}(\mathbb R)$ is an orthogonal matrix ($\det A = \pm 1$) and $v,w\in \mathbb R^3$. I've tried writing ...
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33 views

Alternative matrix representation for translation

The ''usual" way to write translation for $\textbf{v}\in \mathbb{R}^2$ is with the following $3\times3$ matrix $$ \left( {\begin{array}{ccc} 1 & 0 & x_{0} \\ 0 & 1 & y_{0} \\ ...
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1answer
28 views

Proving a linear map is injective

There is a linear map $T:V\rightarrow W$, and $U\subseteq V$ is a subspace such that $$U\cap \ker T=\{0_V\}.$$ I want to prove that the map $T'=T\big|_U:U\rightarrow W$ is also injective. I want to ...
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Formula for connect line between points based on less value of x-axis

I'm stuck to find an equation or formula that connect a straight line to a given n points, based on a less value of x-axis. For example, i have 5 points each has (x,y), want the equation to connect ...
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15 views

Prove that the Conjuguate gradient method converges in at most $n$ iterations

I am trying to probe this corollary in a numerical PDE book: If $A\in \mathbb{R^{n\times n}}$ is symmetric and positive definite, then for some index $m\leq n$ , the residual $r_m$ generated by the ...
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19 views

Sign of Composition of Permutations

Let $\sigma$ be a permutation in $S_5$ with a sign of $-1$. Let $\pi$ be any other arbitrary permutation of $\{1,...,5\}$. What is the sign of the composition $\pi^{-1} \sigma \pi$? Is there any ...
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Decomposing change in estimates

In this scenario, an estimate of a variable is given by the observed data multiplied by a representative weight. Let the observed data at the current time be $d_t$ and the weight $w_t$. The difference ...
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16 views

Isomorphisms with invariant linearly independent dense subset.

If $T$ is an isomorphism acting on a separable Banach space $X$, can we find a countable, dense, linearly independent set $D\subset X$ such that $T(D)=D$? If $X$ is finite dimensional, then the answer ...
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30 views

Help converting currencies of 3 countries (linear algebra problem)

Three countries, $A$, $B$ and $C$ trade goods and services in a closed economy. The percentage of the total production of each country which is consumed by any given country is given in the ...
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17 views

self-adjoint operator over a three dimensional vector space [on hold]

How do I prove that a self-adjoint operator over a three dimensional vector space, is a matrix $$X= \left( \begin{matrix} a & x\\ x^t & B \\ \end{matrix} \right),$$ ...
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45 views

$a-c = \frac{b}{2},\ a-b = \frac{c}{6},\ b+c = 32$ find $a$ =?

I am getting frustrated as I am fighting with this. Please help. I know $$ 2a = b+2c \\ 6a = 6b+c $$ but after this i get confuse what to do next ?
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number of linear maps from $V\to V$

Question is to find total number of linear maps from vector space $\mathbb R^3(\mathbb R)$ to vector space $\mathbb R (\mathbb R)$ which are not Onto? I think trivial map, $T (x, y,z)=0$ could be one ...
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18 views

Find Point on the line segment (7/8) of the way connecting points P and Q

with P = (4,3,-4) and Q = (5,-4,3). My thinking is take the distance between the two, which is (1,-7,7) and taking 7/8 of it which is (-7/8,-49/8,49/8). But I feel like that is wrong and I have to ...
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14 views

Form of Matrix for Reflection about a Line

I've seen a bunch of variations on the wonderful properties of this specific matrix. My textbook gives one algebraic form in particular that I'm having a bit of trouble verifying: Any help here? I ...
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Calculating Normals across a sphere with a wave-like vertex shader

This is a bit of a CS question, but more than not it's a 3D math problem. I've been trying to get the correct normals for a sphere I'm messing with using a vertex shader. The algorithm can be boiled ...
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20 views

product of Matrices notation meaning

I am trying to workout what the meaning of the notation is below. $D$ is a matrix and it is the product from $1$ to $n$. However, the $k(i)$ notation of the matrix and the subsequent $k:(1,2,....n) ...
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Prove that vector space and dual space have same dimension

As an exercise in my textbook, I need to prove that if $V$ is a finite dimensional vector space with dual space $V^*$ over $\mathbb{R}$, then dim$(V)$=dim$(V^*)$. Let $\omega\in V^*$ and let ...
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show that [T]β is a diagonal matrix

$V = P_1(R), T(a + b(x)) = (6a - 6b) + (12a - 11b)x$, and $β = \{3+4x, 2+3x\}$ Show that $[T]β$ is a diagonal matrix I am totally confused about how to write down the matrix form of this ...
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26 views

help with this demonstration of linear algebra [on hold]

V be a vector space over K of finite dimension, T: V* → W a linear transformation. Prove that there exists a unique v∈V such that: T(f)=f(v) for all f∈V*
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SVD of a parametrized matrix.

Suppose we have a parametrized matrix $Z(λ)\in R^{m\times n}$ where $λ\in(a,b)$ and $Ζ(λ)$ is an analytic function of $λ$, e.g. $Z(λ)=λA+(1-λ)B$ where $A,B \in R^{m\times n}$. In general, the ...
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1answer
34 views

Proof help - Linear Transformation [on hold]

$T:V\to W$ is a linear transformation, and $\dim(T\circ T(V))=\dim(T(V))$. Proof $T\circ T(V)=T(V)$.
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Maximizing inner product

Suppose we have two row vectors $a$ and $b$ of nonnegative real numbers such that, for $j<k$ $a_j\leq a_k$ and $b_j\leq b_k$. Let P be a permutation matrix. Can we prove (or disprove) that $$ ...
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31 views

For which values ${(a,1,0),(1,a,1), (0,1,a)}$ form a basis?

To the set be a basis we should have: $$x(a,1,0)+y(1,a,1)+z(0,1,a) = (0,0,0)\implies x=y=z=0$$ so: $$ax + y = 0\\x + ay + z = 0\\y + az = 0$$ which is a system that only has a unique solution if ...
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Is it possible to construct a system of equations for which the set of solutions is a plane?

I may just be overthinking it at this point, but I've struggled to come up with one system of equations that results in a plane. Is it possible? I can't seem to convince myself of a disproof that its ...
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92 views

Show that a given matrix always has an eigenvector in $\mathbb{R}$ Can somebody give a hint?

The given exercise is, for all $\theta$ in $\mathbb{R}$, show that the matrix always has an eigenvector in $\mathbb{R^2}$ $$ A = \begin{pmatrix} \cos(\theta) & \sin(\theta) \\ \sin(\theta) & ...
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406 views

Can the null space be empty?

I was reading a proof of the theorem that the range of a linear map $T$ is always a subspace of the target space, and when the author was showing that the $0$ vector was included in the range, he made ...
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23 views

simple systems of equations problem

Choose h and k such that the system has 1) no solution, 2) a unique solution, and 3) many solutions. Give separate answers for each part. x-3y=1, 2x+hy=k For 1) and 3), isn't that impossible? And ...
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W subspace of $\mathbb{R^4}$ with $\mbox{dim}W = 3$ and $U =\cdots $. Find $\mbox{dim} U+W$ and $\mbox{dim} U\cap W$

Let $W$ be a subspace of $\mathbb{R^4}$ with $\mbox{dim}W = 3$ and $U = \mbox{span}((1,2,1,3),(3,1,-1,4))$. Find $\mbox{dim} U+W$ and $\mbox{dim} U\cap W$ Well, so $\mbox{dim } U = 2$, clearly, ...
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An Optimal Value of a Diagonal Matrix $\Xi$ in $ H = U \Xi$

We have access to very accurate estimates of matrices $H$ and $U$ (both are $n \times k$, $n > k$) such that the following relationship holds $$ H = U \Xi$$ where $\Xi$ is a $k \times k$ diagonal ...
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32 views

Compute the Frobenius norm

I'm trying to compute the Frobenius norm of $L^{-1}$ where $$L^{-1}= I + N + N^2 + ... + N^{n-1}$$ and $N$ is strictly lower triangular. Can anyone suggest some way to do this?
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35 views

Group Action and “nice” Approximation!

Studying the action of a group $G$ on a set $X$ is naturally the same as looking at the group homomorphism $\alpha: G \rightarrow Perm(X)$. So, for a given group $G$, classifying all sets $X$, on ...
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Question about linear operator [on hold]

Let $S: R^2 \rightarrow R^2 $ be a linear operator such that $ S^2 = S$ and $ S\not= 0, S \not= I$. Prove that exists a ordered basis $B$ such that $ [S]_B = \left( \begin{array}{cc} 1 & 0 \\ 0 ...