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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Looking for 'elementary' approach that deals with Hom$_R(\oplus_{i \in I}M_i, N) \overset{\simeq}\to \Pi_{i \in I} \text{Hom}_R(M_i,N)$

I am trying to make this question as clear as possible. I will have to elaborate a bit though in order to do so. I am in a first semester linear Algebra course (although people of a higher semster ...
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491 views

Prove that the largest singular value of a matrix is greater than the largest eigenvalue

Let $\sigma_1$ be the largest singular value of the matrix $A = (a_{ij})$. Show that $\sigma_1 >= \lambda_{max}$, where $\lambda_{max}$ denotes the largest eigenvalue of A, and that $\sigma_1 >=...
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Basis vector find

Let $\alpha_1=[ 2,1,3,0] $ $\alpha_2=[ 1,1,1,-1] $, $\alpha_3=[ 2,-1,5,4] $, $\alpha_4=[ 1,2,0,-3] $, $\alpha_5=[ 3,1,6,1] $ be vectors from $\mathbb{R}^4$ . From vectors system ($\alpha_1,\alpha_2, \...
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137 views

Metric and Convariant Tensor

$g_{ij}$ is the metric tensor. Show that $g^{ij}$ which satsifies $g_{ij}g^{jk}=\delta_i^k$ is a covariant tensor of rank $2$. I am not sure how to show this? Does it instead mean to show that $g_{...
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139 views

Condition number perturbation

I have a matrix of the form $\tilde{H} = H + i A A^\dagger$. It is known that $H$ is hermitian and that $\tilde{H}$ is invertible and $A A^\dagger$ has a kernel of dimension $\geq 1$. I want to study ...
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276 views

Determine if the following function is one-to-one and/or onto

$T(x,y,z) = (xy,yz,xz)$ For one to one, I made $(x,y,z)=(u,v,w)$ and solved. $$xy=uv\to y=\frac{uv}{x}$$ $$\frac{uz}{x}=w$$ $$xz = uw \to x = u$$ $$uy = uv \to y = v$$ $$vz = vw \to z = w$$ So ...
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143 views

Proving solution behavior of a 2x2 system of ODEs with arbitrary real constant coefficients (given trace and determinant conditions)

I have a system of differential equations: x'1 = ax1 + bx2 x'2 = cx1 + dx2 where a, b, c, and d are arbitrary real numbers. I have an iff statement I'm looking to prove: Show that all solutions ...
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21 views

Decide whether vectors are linear independent

We have $\alpha_1, \alpha_2, ...,\alpha_n$ set of $n$ vectors in linear space $V$ such that every $n-1$ subset of vectors is linear independent. The question is if this condition is sufficient to ...
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1answer
47 views

Is the matrix inequality $P > Q \geq 0$ implies $P^2 > Q^2$?

Assume $P$ and $Q$ are positive definite and positive semi-definite, respectively, and both are symmetric. Then, is it true that \begin{equation} P > Q \;\; \Longrightarrow \;\; P^2 > Q^2 \;\;...
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Gram Schmidt Process verification

Use the Gram-Schmidt process to generate an orthogonal set from the given linearly independent vectors: $w_1=\begin{bmatrix} 1 \\ 1 \\ 0 \\ \end{bmatrix} $ $w_2=\begin{bmatrix} 0 \\ 2 \...
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1answer
94 views

Weakly singular integral operator well-defined and bounded

I need to show that the weakly singular integral operator $T_\alpha$ defined by: $$(T_\alpha f)(x)=\int_0^1|x-y|^{-\alpha}f(y)dy$$ where $0<\alpha<1$, is well-defined and bounded on $L^p([0, 1])$...
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87 views

Matrix Factorization Difference

I've just learned about $LDL^T$ decomposition. And i found that there are many other decomposition such as QR decomposition and cholesky decomposition. I don't understand what's the difference between ...
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65 views

Equal multiples of a nonzero vector implies equal factors

$V$ is a vector space with zero element and let $v \in V$. Suppose $av = bv$ and $v \neq 0$. Show that $a = b$. Anyone can guide me on this? Thank you!
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125 views

Addition in linear vector spaces

In the definition of linear vector spaces, one of the axioms is that the addition must be commutative and associative. The addition of scalars and matrices are both commutative and associate. Can ...
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23 views

On functions and their linear independence

How would you access the following problem: Show that the set of functions $$ \phi_n : \mathbb{R}_{>0} \rightarrow \mathbb{R}$$$$\phi_n(x) = \frac{1}{n+x}$$for $n \in \mathbb{Z}^{\ge 0}$ is ...
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4answers
510 views

How is this matrix solved?

for the matrix $$A=\begin{bmatrix} 1 & 0 && 0 \\0 &\cos \frac{\pi}{3} &&\sin\frac{\pi}{3} \\0 & -\sin\frac{\pi}{3}&& \cos\frac{\pi}{3}\end{bmatrix}$$ how it ...
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1answer
61 views

Show that $P^{t}AP$ and $P^{t}BP$ are diagonal matrices

Suppose $A$ and $B$ are symmetric matrices of the same order. If $AB=BA$, Show that $$P^{t}AP$$ and $$P^{t}BP$$ are diagonal matrices. Since $A$ is symmetric, A is diagonalizable. Moreover there ...
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186 views

Proving an isomporphism between all real 2x2 matrix under addition and $R \oplus R \oplus R \oplus R$

Here is my current issue: Let $M$ be the group of all real 2x2 matrices under addition. Let $N=R \oplus R \oplus R \oplus R$ be a group under vector addition. Prove the $M$ and $N$ are isomorphic. I'...
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suppose $S$ is a LI subset of a vector space $V$ and $u$ is a vector in $V$ with $u$ not in $Span(S)$. Show $\{v_1,v_2,…,v_n,u\}$ is LI

suppose $S=\{v_1,v_2,...,v_n\}$ is a linearly independent subset of a vector space $V$ and $u$ is a vector in $V$ with $u$ not in $\operatorname{Span}(S)$. Show $\{v_1,v_2,...,v_n,u\}$ is linearly ...
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1answer
29 views

Find the Fourier coefficients of $cos^2(x)sin^2(x)+2cos^3(x)$

So I know that the Fourier coefficients are expressed as: $$a_0 = \frac{1}{\sqrt{2}\pi} \int_{-\pi}^{\pi}f(t)dt$$ $$c_k = \frac{1}{\pi}\int_{-\pi}^{\pi}f(t)\cos(kt)dt$$ $$b_k = \frac{1}{\pi}\int_{-\...
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164 views

Let $T$ be a self-adjoint operator and$\langle T(w),w\rangle>0$ . If $\operatorname{dim}(W) = k$ then $T$, has at least $k$ positive eigenvalues

Qn: Let T be a finite-dimensional complex inner product space, and T a self-adjoint linear operator. Suppose there exists a subspace W of V such that $\langle T(w),w \rangle$ is positive for all non ...
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116 views

Using my orthonormal basis to find a polynomial that best approximates the polynomial $t^3$

I want to find the second-order polynomial that best approximates $t^3$, with respect to the norm of the vector space $V$. I first proved the bracketing map given in the problem was indeed an inner ...
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162 views

Trace and transpose of a Matrix

I have a recurrence relation as follows $ \left\{ \begin{array}{ll} R_0=H & \mbox{if } n = 0 \\ R_1 =sR_0 \hspace{.1cm} A & \mbox{if }n=1\\ R_{n+2} =\frac{s}{n+2}\{ R_{n+1} \...
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118 views

Orthogonal complement

Let $\{w_1,w_2,...,w_k\}$ be a basis for a subspace $W$ of $V$. Show that $W^⊥$ consists of all vectors in V that are orthogonal to every basis vector. I know that the intersection of the two ...
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1answer
30 views

What does it mean if a function commutes with its adjoint?

What does it mean if a function "commutes with its adjoint"? A matrix, $A$ is self-adjoint if $A = A^*$
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44 views

Orthogonal basis versus non-orthogonal basis

I have a question about the non-orthogonal and orthogonal basis. Let's say we have non-orthogonal independent basis $x_1$, $x_2$, and $x_3$. If I want to represent a vector $x$ using two of them (...
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1answer
31 views

Find matrix that implements Jordan normal form

I have a matrix $$B=\begin{pmatrix}1&2&3\\0&4&5\\0&0&6\end{pmatrix}$$ I have calculated the eigenvectors: $$\{\begin{pmatrix}-1\\0\\0\end{pmatrix},\begin{pmatrix}-\frac{2}{3}\\...
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32 views

Factoring Polynomials and primality testing in R[x]

Say I want to factor a polynomial in R[X], but this polynomial has no roots in R. How exactly do you go about factoring it? For example, if we have $f(x) = x^4 + 16$, this factors into $(x^2 - 2\sqrt{...
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1answer
51 views

Linear Algebra Similarity Orbit

Let $A$ be an $n\times n$ matrix. The similarity orbit of $A$ is the collection of matrices of the form $SAS^{-1}$ where $S$ is an invertible matrix. Describe all the matrices $A$ so that the ...
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40 views

Construct a 2-error correcting Reed-Solomon code over GF(11).

I'am trying to construct a 2-error correcting Reed-Solomon code over GF(11). Cna anyone help me to start?
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64 views

Prove $T:V\rightarrow V$ is orthogonal if and only if $B$ is orthogonal

Suppose a linear transformation $T:V\rightarrow V$ is orthogonal if $||T(f)||=||f||$ for all $f\in V$. Suppose $\mathcal{B}$ is an orthonormal basis for $V$, $T$ is a linear transformation, and $B$ is ...
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1answer
205 views

Characteristic polynomial of 10x10 matrix

Consider the matrix A = $\begin{bmatrix}1&0&1&0&1&0&1&0&1&0&\\0&2&0&2&0&2&0&2&0&2&\end{bmatrix}$. What is the ...
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1answer
96 views

Self-dual code from parity-check matrix

I am trying too make a self-dual code from this parity-check matrix: _ _ | 1 1 1 1 1 1 1 1 | | 1 1 1 1 0 0 0 0 | H= | 1 1 0 0 1 1 0 0 | |1 0 1 0 1 0 1 0| ...
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1answer
37 views

Exact number of vectors in a vector space

I was doing some practice questions in my book and I was asked T/F if there are real vector spaces containing exactly seven vectors. I was thinking that this is True, but the books tells me that the ...
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1answer
54 views

Linear Independence of all rows with leading 1's

For a 5 x 8 matrix A, where it has a leading 1 for every row eg: \begin{pmatrix} 1& 0& 0& 0& 0& 0& 0& 0 \\ 0& 1& 0& 0& 0& 0& 0& 0 \\ 0& 0&...
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1answer
34 views

Basis and Dimension

Let V be the set of all vectors of form $(a, b, c)$ where $a + 2b − c = 0$. Find a basis $B$ for $V$ and find $\dim(V)$. I'm stuck on this one. I know it won't have dimension of 3 because it's easy ...
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142 views

$\cos(nx)=Q_n(\cos(x))$ for polynomial $Q_n$ of degree $n$ [duplicate]

Are there any proofs of this equality online? I'm just looking for something very simply that I can self-verify. My textbook uses the result without a proof, and I want to see what a proof would look ...
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3answers
71 views

Determine whether this set of vectors is linearly independent

For $n \geq 1$ determine if the set $S$ is linearly independent. The set $S$ is these vectors: $$ v_1 \equiv(\,1,2,\ldots,n\,)\,,\quad v_2 \equiv(\,1,2^2,3^2,\ldots,n^2\,)\,,\ldots\, v_n \equiv(\,1,...
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Find $T$ such that $X=TYT^{-1}$

Two matrices $X$ and $Y$ are similar if and only if there is an invertible matrix $T$ such that $X=TYT^{-1}$. Given that $X$ and $Y$ are similar, how would you find a value of $T$?
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Symplectic structures on Hermitian matrices

This is a question taken from Ana Cannas da silva's book on symplectic geometry. Let $\xi\in\mathcal{H}$, the vector space of $n\times n$ hermitian matrix. Define $\omega_{\xi}(X,Y)=i\,\text{trace}([X,...
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If a norm is induced from an inner product, then that inner product is unique

So a norm induced from an inner product satisfies the Parallelogram law: $\|f+g\|^2+\|f-g\|^2=2\|f\|^2+2\|g\|^2$. But why does this mean that the inner product is unique?
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Does this system has a solution?

Solve the system of equations $$\begin{cases}2x_1+x_3+5x_4=0 \\x_1+2x_2-x_3=0\\x_1+x_2+2x_4=0\end{cases}$$ I created the matrix for this system and I found that it has many solution but I'm not sure ...
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304 views

Are two matrices similar iff they have the same Jordan Canonical form?

Are two matrices similar if and only if they have the same Jordan Canonical form? Does the Jordan form have to have ordered eigenvalues? For example, if $\lambda_1$ and $\lambda_2$ are eigenvalues ...
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1answer
27 views

Standard Matrix Transformation

Find the standard matrix for the following composition in $\mathbb{R^2}$: A reflection about the $x$-axis followed by a rotation of $\frac{\pi}{6}$ In my test, I answered the following, but ...
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37 views

Linear Algebra Characteristic Polynomials

Let $p(t) = t^n+a_{n-1}t^{n-1}+a_{n-2}t^{n-2} + \cdots + a_1t+a_0$. Show that the characteristic polynomial of the matrix A below \begin{bmatrix} 0 & 0 & \cdots & & & -a_0\\ 1 &...
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79 views

Any inner product space $V$ is a normed space with norm $||f||=\sqrt{\langle f, f \rangle}$

(1) Show that any inner product space $V$ is a normed space with norm $||f||=\sqrt{\langle f, f \rangle}$, $\forall f\in V$. Can someone please explain what kind of proof is required here? I've ...
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Mathematical explanation for the Repertoire Method

There are a few questions already about this method, which has stumped me for a long while. The process is explained, for instance, here: Repertoire Method Clarification Required ( Concrete ...
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62 views

Linear Algebra Help

How to solve these questions related to Matrix Inverses in Linear Algebra?
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How does Dummit and Foote's abstract algebra text compare to others? [closed]

I am looking for a good book on abstract algebra (and if possible linear algebra). Obviously as most of these texts are fairly expensive I want to know for sure which one is best for me. Could ...
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1answer
55 views

Orthonormal basis implies that the inner product equals the coordinate vectors under the basis multipled together

If $V$ is a finite-dimensional product space, and $B$ is a basis for $V$, then $B$ is orthonormal iff $\langle f,g \rangle=[f]_B\cdot [g]_B$. How can we prove this result in both directions (i.e. one ...