# 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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### A matrix equation involving eigendecomposition

Let $p<n$ and -$H\in\mathbb{R}^{n\times n}$ be symmetric with eigendecompoistion being $H=U\Lambda U^{\text{T}}$, -$A\in\mathbb{R}^{n\times p}$, -$D\in\mathbb{R}^{p\times p}$ be a diagonal ...
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### Variety of submodules of a finitely presented module

Assume that $A$ is a graded $k$-algebra with $A_d=k$ in every degree $d$. Let $M$ be an $A$-module with finite free presentation $F_1\to F_0\to M\to 0$. I want to undertand the collection of finitely ...
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### Finding values for which a linear transformation $L: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ satisfies $v_1, v_2 \in ker(L)$ and $L(v_3) = (1,1,1)$

I am working on an old exam question as preparation for my exam this coming week. The question states that $v_1 = (1, a, a^2), v_2 = (a^2, 1, a), v_3 = (a, a^2, 1)$. Part A asked to find the values ...
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### Significance of this note about definition of determinant in Artin's Algebra

This question is from Artin Algebra, second edition. After defining determinant and proving many of its properties, author comments about possible alternative approach of defining determinant on P23: ...
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### Can i prove that this matrix is PSD?

I have matrix $A \in \mathbb{R}^{N \times N}$, such that $A(i,j)=trace(B_iCB_j), \forall ij$. Matrices $B_i$ and C are PSD and symmetric with positive entries. Can I prove that $A$ is PSD too? In ...
Suppose A is a 3x3 matrix whose columns are orthogonal and the length (two-norm) of each column equals 4. Then what is $A^T*A$? ...