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.

127034 questions
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Is there a geometric proof that the determinant of a 3x3 matrix is invariant under switching rows and columns?

A basic fact about $3$ dimensional vectors is that the quantity $\pm\det\left( \begin{array}{ccc} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \\ \end{array} \right)$ is equal to the volume of the parallelepiped…
Jonah Sinick
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Is there such a thing as "quadratic independence" (and higher generalizations of linear independence)?

The notion of linear independence is very well-known and well-understood. However, is there a way to generalize the definition to other types of independence -- such as perhaps "quadratic independence", "polynomial independence", "harmonic…
user541686
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Reason for thinking of vector as "row" and "column" vectors in linear algebra

Consider the $n$-tuple $(x_1,\ldots,x_n)$ with entries in some field $K$. What is the reason for perceiving this tuple either as a row vector, $$ [x_1,\ldots,x_n]$$or as a column vector…
resu
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How to construct change of basis matrix

How do I construct a change of basis matrix? For example in $\mathbb R^3$, how to construct matrix changing basis from $A$ to $B$? $A=\begin{pmatrix} 1 \\ 0 \\5 \end{pmatrix}\begin{pmatrix} 4 \\ 5 \\5 \end{pmatrix}\begin{pmatrix} 1 \\ 1 \\4…
Josh
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Why Aren't "Similar" Matrices Actually the Same?

In linear algebra, a matrix $B$ is said to be "similar" to $A$ if $B=C^{-1}AC$, that is $B$ = a matrix $A$ multiplied by a third matrix $C$, and its inverse, $C^{-1}$. In regular algebra, if I take a number $x$, and multiply it by $\frac{1}{2}$ and…
Tom Au
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Odd and even functions- a direct sum?

Question: Let $V$ be the vector space of all functions $\Bbb R\to \Bbb R$. Show that $V=U \oplus W$ for $$U=\{f\ | \ f(x)=f(-x)\ \ \forall x\}, \quad W=\{f \ |\ f(x)=-f(-x) \ \ \forall x\}$$ What I did: I did prove that $U \cap W$={$0$}. But proving…
jreing
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How to show that a linear map is surjective?

Sorry if this is somewhat a duplicate. The answers I see deal with functions in general rather than linear maps. Let $T$ be a linear map from $U$ to $V$. I understand that by definition a linear map is injective if every element in the range gets…
Justin
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Show that a positive operator is also hermitian

I'm having a little difficulty with this. Given some positive operator $A$, show that it is also hermitian. (A positive operator is defined as $\langle Ax,x\rangle\ge 0$ for all $x \in V$ where $V$ is some vector space.) Here's what I have so…
randomafk
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Geometry: How to determine if two lines are parallel in 3D based on coordinates of 2 points on each line?

I am a Belgian engineer working on software in C# to provide smart bending solutions to a manufacturer of press brakes. In this context I am searching for the best way to determine if two lines are parallel, based on the following information:…
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Minimal polynominal: geometric meaning

I am currently studying Chapter 6 of Hoffman & Kunze's Linear Algebra which deals with characteristic values and triangulation and diagonalization theorems. The chapter makes heavy use of the concept of the minimal polynomial which it defines as…
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What does a symmetric matrix transformation do, geometrically?

I need some visual intuition behind what exactly a symmetric matrix transformation does. In a $2 \times 2$ and $3 \times 3$ vector space, what are they generally?
JobHunter69
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How to geometrically interpret $\sum^{p}_{1}\lambda_{i}(A)=\operatorname{tr}(A)$?

$A$ is a $p\times p$ real matrix and $\lambda_{i}$ are its eigenvalues. $\operatorname{tr}(A)$ is the trace of $A$. How to geometrically interpret $\sum^{p}_{1}\lambda_{i}(A)=\operatorname{tr}(A)$? I have learnt linear algebra for two semesters. I…
Jill Clover
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The real numbers form a vector space over the rationals (i.e. with Q as the scalar)

How would one go about proving this? I'd like someone to just point in the right direction.
Joris
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What is the difference between diagonalization and orthogonal diagonalization?

I am confused about the following. When you diagonalize a $n\times n$ matrix $A$, you write $A$ as $PDP^{-1}$ with $P$ being orthogonal. Because if $P$ wasn't orthogonal, it wouldn't be invertable. Then why don't we call this "orthogonal…
Edward Stumperd
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does a rectangular matrix have an inverse?

I know all square matrices have easily to identify inverses, but does that continue on with rectangular matrices?
J. Doe
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