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Questions tagged [linear-algebra]

For questions on linear algebra, including vector spaces, linear transformations, systems of linear equations, spanning sets, bases, dimensions and vector subspaces.

Linear algebra is concerned with vector spaces and linear transformations between them:

$$(x_1, \dots, x_n)\to a_1x_n+\dots+a_nx_n$$

Concepts include systems of linear equations, bases, dimensions, subspaces, matrices, determinants, kernels, null spaces, column spaces, traces, eigenvalues and eigenvectors, diagonalization and Jordan normal forms.

This is a general tag, most of the subjects included have secondary tags, e.g.

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Short proof that $X^2 = X \Rightarrow X^{100} = X$

Given that a matrix $X$ satisfies $X^2 = X$ it is clear that $X^{100}=X$ by repeated multiplication of $X$. Algebraically, we might write: $$X^{100} = (X^2)^{50}=X^{50}=(X^2)^{25}=X^{25}=X(X^2)^{12} = \dots = (X^2)^2 = X $$ But this seems like too…
Daniel Pietrobon
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$AX = 0$ and $BX = 0 $ implies A and B are row equivalent

I'm trying to prove that if the systems $AX = 0$ and $BX=0$ are equivalent, then the matrices $A$ and $B$ are row equivalent. Proving the converse was very simple, but this one seems harder. I saw a few answers to similar questions posted here…
Quark
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Understanding a proof from Serge Lang's "Linear Algebra" p. 15

I am working through Serge Lang's undergraduate text: "Linear Algebra" and I've gotten hung up on a particular claim in one of his proofs on p. 15. The result is: Theorem 3.1: Let $V$ be a vector space over the field $K$. Let $\left\{…
dwar
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A problem in Linear algebra

Suppose $A$ is a $n$ by $n$ matrix with entries $a_{ij}$ such that $$|a_{ii}|>\sum_{k\neq i}|a_{ki}|$$ for $i=1,2,...,n$, prove $A$ is invertible.
user5644
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$T: V \to V$ is a linear map. If $\dim(\ker T \cap \text{Im}T)\neq0$, prove $\dim\ker T^2\geq2$

$T: V \to V$ is a linear map. If $\dim(\ker T \cap \text{Im}T)\neq0$, prove $\dim\ker T^2\geq2$ so I can deduce $\dim\ker T\geq 1$ Because the intersection of the image and the kernel, is contained in the kernel. And it is known that $\dim\ker…
Kantig Shoter
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The boundedness of $L_1$ norm $\|(I+A)^{-1}\|_1$ if both $\|A\|_1$ and $\|A^{-1}\|_1$ are bounded

Assume that $A \in \mathbb{R}^{N \times N}$ is a positive semi-definite matrix, both $\|A\|_1$ and $\|A^{-1}\|_1$ are uniformly bounded as $N \to \infty.$ Here $\| \cdot \|_1$ is the induced $L_1$ Norm. Prove that there exists a constant $\kappa >…
Golden Silence
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Eigenvalues and eigenvectors computation (in infinite dimension)

Let $T$ be the backward shift operator: $Tv = T(v_1,v_2,....) = (v_2,v_3,....)$. I would like to determine all the eigenvectors and eigenvalues. So far I have the following: It is evident that $(\alpha, 0,0,....)$ is an eigenvector for the…
newb
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The Definition of Orthogonal Complement

In Linear Algebra Done Right by Axler, the author defines the orthogonal complement as follows: If $U$ is a subset of a vector space $V$, then the orthogonal complement of $U$, denoted by $U^\perp$, is the set of all vectors in $V$ that are…
Koda
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Derivation linear map examples

From Humphreys' Introduction to Lie Algebras and Representation Theory: By an $F$-algebra (not necessarily associative) we simply mean a vector space $U$ over $F$ endowed with a bilinear operation $U\times U\rightarrow U$, usually denoted by…
PJ Miller
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Orthogonal matrix confusion

I have a confusion about orthogonal matrix. If columns of a square matrix are orthonormal to each other, is the matrix orthogonal? If yes, then are the rows of the matrix also orthonormal? Why? Why is it that QQ'=I? I get Q'Q=I but why QQ' is also…
Tom Bennett
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How can one convert rational matrices into integer ones?

Let $G$ be a finite subgroup of $GL_n(\Bbb{Q})$. I want to show the existence of a matrix $A\in GL_n(\Bbb{Q})$ with the property that $AGA^{-1} \subseteq M_n(\Bbb{Z})$.
kian
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$T$ is normal if and only if exist polynomial $p$ s.t $T^{*}=p(T)$

Let $V$ be an inner product space over $\mathbb{C}$, and let $T:v\to V$ be a linear transformation. Need proving that $T$ is normal if and only if exist polynomial $p\in\mathbb{C}[x]$ s.t $T^{*}=p(T)$. Thanks!
17SI.34SA
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For self-adjoint operators, eigenvectors that correspond to distinct eigenvalues are orthogonal

So I was looking for a proof for the next theorem. $V$ is inner product space $T: V\rightarrow V$ self adjoint linear map. $ \lambda_{1},\lambda_{2} \in \mathbb{F}$ so that $ \lambda_{1} \neq \lambda_{2}$ $ v_{1},v_{2} \in V$ so that $ 0_{v} \neq…
wantToLearn
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How do I find the determinant of this matrix?

I'm preparing for an exam currently, and I came across this question: I have noticed that A can be constructed from the matrix on the left by a series of row operations, so I had the idea maybe to express A as a product of elementary matrices as…
HarveyR
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Find a basis for the range and kernel of $T$.

Find a basis for the range and kernel of $T$. $$A =\begin{bmatrix} 2 & 0 & -1\\ 4 & 0 & -2\\ 0 & 0 & 0 \end{bmatrix} $$ Attempt at Solving for Basis of Range: On finding the basis for the range, I know that the range is the same thing as the…
briteId
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