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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Expressing the product Ax as a linear combination of the column vectors of A

Expressing the product Ax as a linear combination of the column vectors of $A$= $\begin{bmatrix} 4 & 0 & -1\\ 3 & 6 & 2\\ 0 & -1 & 4 \end{bmatrix}$ $\vec{x}$=$\begin{bmatrix} -2\\ 3\\ 5 \end{bmatrix}$ I get it now. They just want me to…
user8479
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Is the function T $\mathbb R$-linear?

Let $T:\mathbb R^2\to \mathbb R^2$ be a mapping such that $T(C)$ is a convex set in $\mathbb R^2$ whenever $C$ is convex set in $\mathbb R^2$ and $T(0,0)=(0,0).$ Is $T$ $\mathbb R$-linear? We have to show here that $T(ax+by)=aT(x)+bT(y)$ for all…
Mini_me
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Sum of vector subspaces

I am trying to self study linear algebra and am stuck on a problem. It comes from Axler's Linear Algebra done right example 1.38 and I don't understand the solution that I could find online. Suppose that $U=\{(x,x,y,y)\in F^4 : x,y \in F\}$ and…
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Prove that three $2\times2$ matrices that commute are linearly dependent

Statement: Suppose that $A$, $B$ and $C$ are complex $2\times2$ matrices, any two of which commute under matrix multiplication. Show that $A$, $B$ and $C$ are linearly dependent. I think one method is to show the existence of $a,b,c\in\mathbb C$,…
Frenzy Li
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Is a projector matrix the inverse of itself?

I want to confirm if a projector matrix is its own inverse. I have $x=Px$ and $Px=P^2x$, so premultiplying the second equation with $P^{-1}$ twice, I get $P^{-1}x=Px$ for all x, implying $P^{-1}=P$. Is this reasoning correct? So are all projection…
Bravo
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What's the point of duality?

I'm taking a second course in linear algebra. Duality was discussed in the early part of the course. But I don't see any significance of it. It seems to be an isolated topic, and it hasn't been mentioned anymore. So what's exactly the point of…
mike
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Find an equation of the plane passing through 2 points and perpendicular to another plane

Find an equation of the plane that passes through the points $(0-2,5)$ and $(-1,3,1)$ and is perpendicular to the plane $2z = 5x + 4y$. Here's what I have so far: The plane through $(0,-2,5)$ is $ax + b(y+z) + c(z-5) = 0$. And the plane also…
minhaz1
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The Determinant of an Operator Equals the Product of Determinants of its Restrictions

Consider the following definitions and proved theorems Definitions $1$. The field $\Bbb{F}$ is $\Bbb{R}$ or $\Bbb{C}$. $2$. $V$ is a vector space over $\Bbb{F}$. $3$. $\mathcal{L}(V)$ is the vector space of all operators on $V$ (that is, linear…
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Why vectors in Linear Algebra start from point (0,0)?

I learn linear algebra in university and I was wondering why vectors in linear algebra always start from the point $(0,0)$? how many kinds of mathematical vectors out there? Is it legit to use other kind of vectors in linear algebra apart from that…
LiziPizi
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Visualization of the dual space of a vector space

I am wondering what the motivation was for defining a dual space of a vector space, and how to visualize the dual space. I'm asking since it doesn't seem to me to be intuitive to deal with such a space. In particular, I'm looking for questions…
fdzsfhaS
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Matrices whose Linear Combinations are All Singular

I'd like to know if the following problem of elementary linear algebra is already solved / solvable. For two (singular) $n\times n$ matrices $P$ and $Q$, if $\det(\lambda P+\mu Q)=0$ for any $\lambda,\mu\in\mathbb{R}$, what are conditions on $P$ and…
finnlim
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Does non-degenerate and positive semi-definite imply positive definite?

Let $V$ be a finite-dimensional real vector space and $$B:V\times V\to\Bbb R$$ a bilinear form. Suppose that $B$ is non-degenerate and positive semi-definite, i.e. $B(X,Y)=0,\forall Y\in V\implies X=0$ $B(X,X)\ge0,\forall X\in V$. Does that imply…
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help understanding a paragraph from Linear Algebra Done Right

I am attempting to work my way through the 3rd edition of "Linear Algebra Done Right", but there's a paragraph on page 14 that I don't understand. I have struggled with it myself for a few hours and have come to the conclusion that I need some…
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how to find maximal linearly independent subsets

Given a set of vectors, we can compute the number of independent vectors by calculating the rank of the set, but my question is how to find a maximal linearly independent subset. Thanks!
chaohuang
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Help finding the determinant of a 4x4 matrix?

Sorry for the lack of notation but the work should be easy to follow if you know what you are doing. Okay my problem is that the book says it can be done by expanding across any column or row but the only way to get what the book does in their…
K. Gibson
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