# 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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### When is “$\Re(\lambda) \gt 0$ for $\lambda \in \sigma(A),A \in \mathbb{R}^n$” true?

Let $A \in \mathbb{R}^{n \times n}$ and $\sigma(A)$ the spectrum of $A$. I am searching for a fast way to check whether $\Re(\lambda) \gt 0$ for all $\lambda \in A$. If $A = A^t$, one only has to ...
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### Vector space generated

Let $(L,+)<(\mathbb R^n,+)$ be a additive subgroup and let $\{v_1,\ldots,v_m\}$ be a maximal linearly independent subset of $L$. Let $V$ be the subspace spanned by $\{v_1,\ldots,v_m\}$. Asumme that ...
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### Check if set of functions is a basis of space

Let $f_a \in R^R$ be function given by $f_a(x)=1$ if $x=a$ and $f_a(x)=0$ if $x \neq a$ for $a \in R$ Decide if set of functions $f_a$ is a basis of space of functions $R^R$ ? I think I know how to ...
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### If det(A) is zero, what is det(adj(A))?

I wanted to prove that det(adj(A))=det(A)^n-1 for an nxn matrix A. I separate the proof in two cases: singular and non-singular matrix A. For the non-invertible A, det(A)=0. In my head, I know that ...
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### Solution of system of three variables

On solving $$2x - 4y + z = 0$$ $$x + y - 4z = 0$$ $$x - y - z = 0$$ I get $$y = 0.6 x$$ $$z = 0.4 x$$ I thought that there was a rule of thumb that you need as many independent ...
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### Evaluating a determinant for eigenvalues

I need to evaluate $$\left| {\matrix{ {3 - \lambda } & 1 & 1 \cr 2 & {4 - \lambda } & 2 \cr 1 & 1 & {3 - \lambda } \cr } } \right|$$ A direct computation ...
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### How come that HSL can contain more information than RGB?

I have noticed weird thing when working with HSL - unlike RGB, it has some blind spots where certain value just does not matter. I'm sure we were taught about this when I had Linear algebra lectures ...
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### Generators of group of “unitary” matrices over a finite field

This is about a group related to $U(n,q)$ and $SU(n,q)$. I know from multiple sources the generators for these groups, but $U(n,q)$ is defined to be the group of matrices $A$ such that $A^*JA = J$ ...
Assume that $A\in \mathbb{C}^{n\times n}$. The $\epsilon-$pseudospectrum of $A$ is defined by $$\sigma_{\epsilon}(A)=\{z\in C \quad | \quad \Arrowvert (zI-A)^{-1} \Arrowvert>\frac{1}{\epsilon}\}.$$ ...