# 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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### Prove that $\text{det}(A)=p_1p_2-ba={bf(a)-af(b)\over b-a}$

Let $f(x)=(p_1-x)\cdots (p_n-x)$ $p_1,...p_n\in \mathbb R$ and let $a,b\in \mathbb R$ such that $a\neq b$ Prove that $\text{det} A={bf(a)-af(b)\over b-a}$ where $A$ is the matrix: ...
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### Systems of linear equations in the same modulus

I am working with a system of linear equations all taken with the same modulus, $n$, there is no assumption on $n$ other then it is at least 3 (really don't want to assume it is prime) I don't have ...
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### Express eigenvectors of $A^{-1}$ in terms of eigenvectors of $A$

I know the eigenvalues of the matrix $A^{-1}$ are $\frac{1}{\lambda_n}$ where $\lambda_n$ are the eigenvalues of $A$. I didn't know their eigenvectors were related; in what way are they related? Also ...
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### Degrees of freedom in a $n \times n$ table

Suppose we have an $n \times n$ table where each row and each column sums to some number $k$. Say that the elements of the table and $k$ are real numbers. Now the question is how many places can we ...
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### finding inner product

This is from my textbook: I don't know how to tell whether the spanning set are actually orthogonal. The textbook's solution is like this, forexample, to see if $P_0(t)$ and $P_1(t)$ are orthognal, ...
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### Interval bounds for symmetric doubly-stochastic matrices (designed with Metropolis weights).

I'm facing an unusual problem with doubly-stochastic matrices, in the context of some undirected graph. I assume that it is connected, but this is not so important for this problem. Let me introduce ...
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### PTM using Hastings-metropolis [on hold]

[Compute the 4 × 4 PTM (pij ) under the T = 2 dynamics of Hastings–Metropolis][1]
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### Find the possible equations of planes containing a line

If I had a line in $R^3$ that had all its points described by $P =\begin{pmatrix} a\\ b\\ c \end{pmatrix}+\lambda \begin{pmatrix} x\\ y\\z \end{pmatrix}$ where $a, b,$ and $c$ are constants, what ...
### Does $B^2 \leq A^2$ imply $\| A^{-1} B\| \leq 1$ for the operator norm?
Assume we have two $n \times n$ real symmetric matrices $A^2$ and $B^2$, such that it holds for some $0\leq\rho<1$ $$0 < (1-\rho)B^2 \leq A^2 \leq (1+\rho)B^2,$$ where "$\leq$" means ...