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Questions tagged [linear-algebra]

For questions about 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.

Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them (e.g. $(x_1, \dots, x_n)\mapsto a_1x_1 + \dots + a_nx_n$). Concepts include systems of linear equations, bases, dimensions, subspaces, matrices, determinants, kernels, null spaces, column spaces, traces, eigenvalues and eigenvectors, diagonalization, Jordan normal forms, and so forth.

This is a general tag, and most of the subjects included in its scope also have secondary tags (e.g. , , , , , etc.). Please consider using the appropriate secondary tags as they are applicable.

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116358 questions
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Examples for proof of geometric vs. algebraic multiplicity

Here you see a supposedly easy proof of a well-known theorem in linear algebra: Although I know I should understand this, I don't :-( Obviously there are too many indices and stuff, so I don't see the forest for the trees. Please could anyone give…
vonjd
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Determinant game - winning strategy

I came across this problem while looking at Putnam problems a while ago: Alan and Barbara play a game in which they take turns filling entries of an initially empty $2008 \times 2008$ array. Alan plays first. At each turn, a player chooses a real…
Evan
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Calculate the determinant of $a_{ij} = \frac{(1+x)^{i+j-1}-1}{i+j-1}$

There is a question asked by my classmate. Looking forward to some ideas, thanks. Set $A=\{a_{ij}\}_{n\times n}$, where $$a_{ij}=\frac{(1+x)^{i+j-1}-1}{i+j-1}.$$ Prove that $\det A=cx^{n^2}$ for some $c$. I have tried to calculate it, but failed.…
ling
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Confusion concerning Lemma 1.12 in Wiles's proof of Fermat's Last Theorem

Let $k$ be a finite field of characteristic $p\neq 2$ (in fact, one only needs to consider the case $p\in\{3,5\}$), let $\Sigma$ be a finite set of primes containing $\infty$ and $p$, and $$\rho_{0}:{\rm…
The Thin Whistler
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Proof Complex positive definite => self-adjoint

I am looking for a proof of the theorem that says: A is a complex positive definite endomorphism and therefore is A self-adjoint. Does anybody of you know how to do this?
user66906
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Structure on $\mathbb{R}$ such that homomorphisms $\mathbb{R} \to \mathbb{R}$ are exactly polynomials?

If we consider $\mathbb{R}$ as a vector space over the field $\mathbb{R}$, then the maps $\mathbb{R} \to \mathbb{R}$ preserving this structure are exactly the linear maps $x \mapsto ax$. In contrast, if we think of $\mathbb{R}$ as an affine space…
Jordan Mitchell Barrett
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Determinant of a Pascal Matrix, sort of

Let $A_{n}$ be the $(n+1) \times(n+1)$ matrix with coefficients $$ a_{i j}={i+j \choose i} $$ (binomial coefficients), where the rows and columns are indexed by the numbers from 0 to $n$ are indexed. Now I want to determine the Determinant and with…
calculatormathematical
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$A$ is some fixed matrix. Let $U(B)=AB-BA$. If $A$ is diagonalizable then so is $U$?

This is from Hoffman and Kunze 6.4.13. I am studying for an exam and trying to solve some problems in Hoffman and Kunze. Here is the question. Let $V$ be the space of $n\times n$ matrices over a field $F$. Let $A$ be a fixed matrix in $V$. Let $T$…
Cousin Dupree
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An elementary way to show that the determinant is non zero

Show that the determinant of the matrix \begin{pmatrix} a && -c && -b \\ b && a - 2c && -c -2b \\ c && b && a -2c \end{pmatrix} is non zero for all integers $a,b,c$ where $abc \ne 0$ There is an interesting way to do this by using integral…
Infinity_hunter
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Proof and Intuition of the Determinant Formula?

[I did notice similar questions were asked here before, but I couldn't find a satisfactory answer for me to grasp as a beginner, so I chose to post this question] I'm just starting to teach myself linear algebra with Linear Algebra and Group…
greenidiot
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Singular value decomposition of positive definite matrix

Let $A$ be a positive definite matrix, and let $A = U \Sigma V^*$ be its singular value decomposition (SVD). Show that $U=V$. What I have done: $A$ is Hermitian, so $A$ is unitarily diagonalizable, say, $A=WDW^*$ where $D$ consists of the…
Gobi
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Product of nilpotent matrices.

Let $A$ and $B$ be $n \times n$ complex matrices and let $[A,B] = AB - BA$. Question: If $A , B$ and $[A,B]$ are all nilpotent matrices, is it necessarily true that $\operatorname{trace}(AB) = 0$? If,in fact, $[A,B] = 0$, then we can take $A$…
user2052
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Avoid the planes - the geometry of grassmannians

Suppose we have $n$ planes $H_1, \ldots, H_n$ in $\mathbb{R}^m$ of codimension $q$, or equivalently of dimension $d=m-q$. I want to choose a vector which does not belong to the planes in a continuous way. There are two versions of this problem,…
Andrea Marino
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Size of a linear image of a cube in $\mathbb{Z}^d$

Suppose that we have an element $v = (v_1, \dots, v_d) \in \mathbb{Z}^d$ such that $\gcd(v_1, \dots, v_d) = 1$. Then $v$ is contained in some base of $\mathbb{Z}^d$ (seen as a free-abelian group or a free module over $\mathbb{Z}$). In particular,…
Michal Ferov
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Prove that matrices of this form have eigenvalues $0,1,\ldots , n-1$

Fix arbitrary real numbers $x_1,\ldots ,x_n$ which are pairwise distinct, i.e. so that $x_i \neq x_j$ for any pair $i \neq j$. Let $A = (a_{ij})$ be the following $n \times n$ matrix: Its diagonal entries are given by the…
baked goods
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