# Questions tagged [linear-algebra]

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 concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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### If $V$ is a vector space, $W$ and $U$ are subspaces of $V$, why is $U^0, W^0 \supseteq ( U \cap W)^0$ true?

If $\textsf V$ is a vector space, $\textsf W$ and $\textsf U$ are subspaces of $\textsf V$, why is $\textsf U^0, \textsf W^0 \supseteq (\textsf U \cap \textsf W)^0$ true? My mind tells me that the ...
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### Dimension of the space of matrices which is commutative to a given matrix.

Suppose I have a matrix $A$ in the space $V$ of $n$ by $n$ matrices. Then it is quite clear that $S=\{B : AB=BA\}$ form a subspace. I want to find out its dimension. I think it depends on the ...
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### Proof that equation $Lf=u$ guarantees satisfiability of weak form $\langle Lf,v \rangle = \langle u ,v \rangle$

Problem Let's observe in a closed interval $[a,b] \subset \mathbb{R}$ real-valued and continuous vectorspace $\mathcal{F}([a,b],\mathbb{R})$. Where $\langle ., .\rangle$ is some scalar product. This ...
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### To prove a matrix is PSD

This question rises from the proof of the outer product Cholesky Factorization. If the matrix $$M=\begin{pmatrix} \alpha&\vec{q}^T \\ \vec{q}&N \end{pmatrix}$$ is positive semidefinite with ...
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### Representing a Bilinear Form in a Matrix

Let $b : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a bilinear form, and $\langle,\rangle$ be the standard inner product of the Euclidean space and $e_1,...e_n$ be the standard basis for the ...
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### Prove the existence of left and right inverse?

A theorem states the following Let $\mathbf{A}$ be a $m \times n$ matrix such that rank $\mathbf{A} = r$. (a) $\mathbf{A}$ has a right-inverse if and only if and only if $r = m$ and $m \leq n$ (b) \$...