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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Does there exist a continous function $f(t)$ on $[0,1]$ for which $\int_0^1 t^3 f(t) dt = 0$?

Does there exist a continous function $f(t)$ on $[0,1]$ for which $\int_0^1 t^3 f(t) dt = 0$? Or can you provide a proof otherwise?
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2answers
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Epimorphism of linear transformation

Find values of parameter t for which transformation is epimorphic: $\psi([x_1,x_2,x_3,x_4])=[x_1+x_2+x_3+2x_4,x_1+tx_2+x_3+3x_4,2x_1+x_2+tx_3+3x_4] $ When this transformation is epimorphic i.e. what ...
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1answer
15 views

Eigenvalue of altered matrix: $pI_n + qA$

As a part of an exercise I have to prove the following: Let $p,q \in \mathbb{R}$. Let $A$ be an $(n \times n)$ matrix. Let $I_n$ be the $(n \times n)$ identity matrix. If $A$ has an eigenvalue ...
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1answer
9 views

Show that $trv=\lim_{t\to 0}\frac{\det(I+tv)-1}{t}$ for any n by n matrix

Prove that for any n by n real matrix $v\in {\mathbb R}^{n\times n}$, $trv=\lim_{t\to 0}\frac{\det(I+tv)-1}{t}$, where $t\in\mathbb R$, $I$ is the identity matirx, and $trv$ denotes the trace of ...
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6answers
224 views

How to solve these equations for x and y..

equations are $(x-y)(x+2y)(2x+y) = 20$ and $x^2+xy+y^2 = 7$ i want the METHOD not the solutions
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2answers
331 views

Given a $4\times 4$ symmetric matrix, is there an efficient way to find its eigenvalues and diagonalize it?

I have a $4\times 4$ matrix $$A=\left(\begin{array}{cccc}8 & 11 & 4 & 3\\11 & 12 & 4 & 7\\4 & 4 & 7 & 12\\3 & 7 & 12 & 17\end{array}\right).$$ I want to ...
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50 views

Two linear maps which commute

If $S$ and $T$ are linear maps over a finite dimensional complex vector space. They commute i.e $ST=TS$. Is there any common subspace under which both are invariant ? I guess it is ...
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1answer
14 views

Determining transformation matrix from six points

Given that I have the locations of three points: p1 = [1.0,1.0,1.0] p2 = [1.0,2.0,1.0] p3 = [1.0,1.0,2.0] ...and I know their transformed counterparts: ...
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0answers
10 views

Lagrange Newton Method Singular Matrix

i implemented the lagrange-newton method in python to find the problem to nonlinear optimizing problem for learning purposes. But every guess i made a guess for the initial values the resulting ...
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0answers
16 views

A basis that simultaniously diagonalizes two matrices?

Given matrices $A$ and $B$, assuming they can be diagonalized with the same S, so that $D_A = SAS^{-1}$ and $D_B = SBS^{-1}$ ... how would one find the basis that makes up $S^{-1}$? I've got two ...
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12answers
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What are some applications of elementary linear algebra outside of math?

I'm TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I'm looking for applications of elementary linear algebra outside ...
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5answers
70 views

What is the motivation/application of dual spaces and transposes?

I've always been baffled as to where transposes come from. I found this question, but the answer isn't satisfying to me - the idea seems to be "dual spaces are important, and you can define transposes ...
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1answer
12 views

How to intersect row and column sub-spaces?

What is the connection or, intersection between row space and column space of a square matrix? how can I intersect two different sub-spaces?
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1answer
325 views

Column and Row Picture for Singular System of 100 Equations (Strang P55, 2.2.32)

Start with 100 equations $\color{#8F00FF}{A}\mathbf{x} = \mathbf{0}$ for $\mathbf{x} = (x_1, ..., x_{1oo})$. Suppose elimination reduces the 100th equation to $0 = 0$, so the system is "singular". ...
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0answers
8 views

Interpolating transformation matrices

I read not to interpolate transformation matrices by linearly interpolating. Can someone explain to me why interpolating transformation matrices by linearly interpolating the matrix components is a ...
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1answer
16 views

Algorithm for the Hill cipher (finding the inverse of the determinant of a $2 \times 2$ matrix modulo $26$)

I have a good understanding of how to do the Hill cipher on paper but putting it into program form is somewhat of a problem. Finding the the determinant is the thing I'm having problem with. On ...
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1answer
36 views

Find a minimal spanning set of a set of matrices

I'm supposed to find a minimal spanning set of $W = \{A \in M_n(\mathbb{R}) | \operatorname{Tr}(A) = 0\}$ First of all, what is a minimal spanning set? I can't find the term anywhere in the notes my ...
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1answer
696 views

Compute Left Eigenvectors

How does one compute the left eigenvectors of a matrix? I cannot seem to quite get the answer.. I don't care what the matrix is. Let's just say I have matrix $A$ and have found the 'right' ...
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1answer
40 views

Describe the following set

We are supposed to describe the set $\bigcup_{n=1}^\infty A_n$ with a proof. $A_n = \{(x, y) \in \mathbb{R}^2 | y-x^{2n} \geq 0 \}$ I get that the set contains all functions where $y\geq x^{2n}$ ...
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0answers
20 views

Is the ratio trace problem convex?

I have a ratio trace problem described as follows: $\arg\max_{w} trace((w^tAw)*inv(w^tBw))$, where A and B are full rank matrices. This problem can be solved via generalized eigenvalue problem. ...
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1answer
162 views

Find basis of the annihilator set

$V$ $= \text{span}\{(1,2,3),(1,1,1)\}$ $\subseteq \mathbb{R}^3$. Find the vectors spanning $V^0$ in terms of the usual basis for $(\mathbb{R}^3)^*$. So we want linear functionals $f \in V^*$ such ...
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1answer
27 views

Faulty proof that $V=U_1 \oplus W$ and $V=U_2 \oplus W$ implies $U_1 = U_2$

The question is as follows: Prove or give a counterexample: if $\ U_1, U_2, W$ are subspaces of $V$ such that $V=U_1 \oplus W$ and $\ V = U_2 \oplus W$, then $\ U_1 = U_2$. I happily ...
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0answers
24 views

Geometrical interpretation of an overidentified linear system

In my econometrics class we talked about Instrumental Variables. Suppose one has a $n\times k$ matrix $X$ of regressors and a $n\times m$ matrix $Z$ of instrumental variables. Given the matrices are ...
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0answers
33 views

Orthogonal vectors to Bivector

If we have set of orthogonal vectors (X) and form that set we create a set of bivectors (Y). Is there a relation among Y based on X? Can we say that Y's are orthogonal as well?
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2answers
50 views

Prove there exist an isomorphism

Let $V$ be a vector space and $U,W,Z$ are subspaces of $V$, where $V=Z \oplus W=Z \oplus U$ Prove there exist linear isomorphism $f:V \to V$ such that for every $\gamma \in Z, \ \ f(\gamma)=\gamma$ ...
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2answers
57 views

Show that a basis change is a linear transformation [on hold]

Show that the map v = [v]E → [v]B for all v ∈ Rn defines a linear transformation TB : Rn → Rn. B = {b1, b2, b3,...,bn} is a basis of Rn. Any vector v ∈ Rn can be uniquely expressed as a linear ...
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1answer
18 views

Linear composition

can you help me with this quest? About composition $f$ and vector space $\mathbf{V}=\mathbb{Z^4_2}$ we know the following: $f \circ f = id_V$,$~~f $ $ \left(\begin{array}{ccc} 1\\ 0\\ 1\\ 0\\ ...
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0answers
34 views

equivalent inner-product vector for one

I have a map that projects a $k$ dimensional vector $x$ to an $m$ dimensional vector $\phi(x)$. The vector function (map) $\phi$ can be any linear or non-linear function of $x$, which is not ...
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2answers
23 views

Prove vectors create a basis

Let $V$ be a vector space and $U,W,Z$ be it's subspaces where $V=Z \oplus U=Z\oplus W$. We know that $\beta_1,...,\beta_k$ is a basis of $U$ and $\beta_i=\gamma_i+\delta_i$ where $\gamma_i \in Z$ and ...
2
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1answer
55 views

Surface normal to point on displaced sphere

I want to calculate the surface normal to a point on a deformed sphere. The surface of the sphere is displaced along its (original) normals by a function $f(\vec x)$. In mathematical terms: Let ...
7
votes
4answers
650 views

A faster way of calculating this determinant?

I'm doing a problem involving Cramer's rule, and one of the determinants I have to work with is as follows: \begin{vmatrix} 1&1&1\\ a&b&c\\ a^3&b^3&c^3 \end{vmatrix} So I ...
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1answer
27 views

Prove that the set of points that make up the unit circle are uncountable

My math teacher asked us to prove that the set of points that make up the unit circle are uncountable. We are supposed to do this by "exhibiting" (not sure if this means it can be a proven through ...
0
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2answers
29 views

Let $V=\{f \in X \mid f(0)=f(1)=0\}$ be a linear subspace of $C[0,1]$. Show $(V,\|\cdot\|_\infty)$ is Banach.

Can you please confirm if my proof is correct and if not show where I went wrong. Thanks! Let ${f_n}$ be a Cauchy sequence in $V$ then $f_n(x)$ is a real number for each $x\in [0,1]$ Hence ${f_n(x)}$ ...
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0answers
10 views

Show that there exists a Hermitian form of signature $(p,q)$.

Let $K = \mathbb{Q}(\sqrt{-2})$ with $V_K = K^n$ considered as a $K$-vector space. Suppose $p, q \in \mathbb{Z}_{>0}$ such that $p + q = n$. Show that for any such $p$ and $q$ there is a Hermitian ...
4
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1answer
43 views

General linear group and special linear group

Consider the general linear group $$GL(n,\mathbb R)=\{g\in {\mathbb R}^{n\times n}\mid\det(g)\neq 0\}$$ Prove that the derivative of the function $f=\det:{\mathbb R}^{n\times n}\to\mathbb R$ is ...
1
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0answers
18 views

Give the following linear transformation find values of parameter

Find values of parameter t for which transformation is epimorphic: $\psi([x_1,x_2,x_3,x_4])=x_1+x_2+x_3+2x_4,x_1+tx_2+x_3+3x_4,2x_1+x_2+tx_3+3x_4 $ When this transformation is epimorphic i.e. what ...
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4answers
42 views

Proving something is a linear transformation? [on hold]

If $T:\Bbb R^3 \to \Bbb R^2$ given by $$ T\begin{bmatrix} a\\ c\\ e\\ \end{bmatrix}= \begin{bmatrix} c\\ a\\ \end{bmatrix} $$ then $T$ is a linear transformation: true or false, ...
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1answer
21 views

Examples of Unitary Matrices with coefficients all having the same amplitude

I am looking for examples of unitary matrices like this one $$A = \frac{1}{\sqrt{2}}\left( \begin{array}{rr} 1 & 1 \\ 1 &-1 \end{array} \right)$$ where each coefficient has the same amplitude, ...
0
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1answer
21 views

The convergence of $f_n(t)$ to $f_n$ in the supremum norm implies $f_n(t)\rightarrow f(t)$ as $n\rightarrow\infty$?

I'm awful at these problems so I was just posting this to confirm whether my solution is correct. As: $$(f_n)_{n\geq 1} \in X \rightarrow f \in X \mbox{ in the supremum norm}$$ $$\mbox{For all } ...
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1answer
26 views

Count and description of vertices of certain faces of the Tridiagonal Birkhoff polytope $\Omega^t_{d+k}$

For $k \ge 1$, $d \ge 2$ and $k \le d - 1$, let ${}^f_d\Omega^t_{d+k} (d;c_k(d - 1))$ be the intersection of $k - 1$ facets of the Tridiagonal Birkhoff polytope $\Omega^t_{d+k}$ with equations: ...
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0answers
16 views

Phase Portrait of DE's

How would I graph the phase portrait of $$ x' = x^2+y^2-2 \qquad y' = y-x^2 $$ ? Could someone provide some insight by hand or perhaps a computer-generated image?
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2answers
40 views

let A be a $2\times 2$ matrix . Then the smallest number $n\in \mathbb N$ such that $A^n=I$ is

let A be a $2\times 2$ matrix $\begin{pmatrix} \sin \frac \pi {18} & -\sin \frac {4\pi} {9}\\ \sin \frac {4\pi} {9}&\sin \frac \pi {18}\end{pmatrix}$. Then the smallest number $n\in \mathbb N$ ...
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9answers
958 views

Why do determinants have their particular form?

I know that for a matrix $A$, if $\det(A)=0$ then the matrix does not have an inverse, and hence the associated system of equations does not have a unique solution. However, why do the determinant ...
2
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4answers
82 views

Is every basis of a finite-dimensional vector space orthonormal with respect to some inner product?

Given a real or complex vector space $V$ and a (finite) basis $B$ of it, does it always exist an inner product on $V$ such that $B$ is an orthonormal basis with respect to it? The question is ...
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0answers
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$A$ be a $10\times 10$ matrix over $\mathbb R$ such that sum of each row is $1$. [on hold]

Let $A$ be an invertible $10\times10$ matrix over $\mathbb R$ such that sum of each row is $1.$ Then which option is correct? A. The sum of the entries of each row of the inverse of $A$ is ...
2
votes
1answer
90 views

How find this invertible matrix $C=\begin{bmatrix} A&B\\ B^T&0 \end{bmatrix}$

let matrix $A_{n\times n}$,and $\det(A)>0$, and the matrix $B_{n\times m}$,and such $rank(B)=m$,and let $$C=\begin{bmatrix} A&B\\ B^T&0 \end{bmatrix}$$ Find this Invertible matrix ...
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1answer
31 views

Solving a linear optimization problem with products and work benches

I am taking a linear algebra course and I have a homework assignment of: A factory produces 5 products T1, T2, T3, T4, T5. Products are made on 3 different work benches P1, P2, P3, which can be used ...
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3answers
34 views

Abstract Linear Transformation Question

I had this question on a quiz today and no idea how to solve it. Please help. Let $ T: \mathbb{R}^n\rightarrow \mathbb{R}^n $ a linear transformation defined by: $\forall \begin{bmatrix}x_1\\x_2\\ ...
0
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0answers
14 views

How can we drive frome the matrix charaterization to affine subspace formula?

My question is about the projections, and it consists of two parts: The projection of any point $x\in\mathbb{R}^n$onto any subspace $V$ of a finite dimensional space is defined by ...
2
votes
1answer
28 views

Unit close disc to prove a matrix algebra identity?

I need to prove that every $3 \times 3$ matrix with real positive entries has one eigenvector with a positive eigenvalue. Now, how do I prove this using the fact that the set $B=\{x=(x_1,x_2,x_3)\in ...