# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...