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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.

127034 questions
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Does the rank-nullity theorem hold for infinite dimensional $V$?

The rank nullity theorem states that for vector spaces $V$ and $W$ with $V$ finite dimensional, and $T: V \to W$ a linear map, $$\dim V = \dim \ker T + \dim \operatorname{im} T.$$ Does this hold for infinite dimensional $V$? According to this, the…
user142870
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How many k-dimensional subspaces there are in n-dimensional vector space over $\mathbb F_p$?

I am asked to find how many there are $k$-dimensional subspaces in vector space $V$ over $\mathbb F_p$, $\dim V = n$. My attempt: 1) Let's find a total number of elements in $V$: assume that $\{v_1, v_2, \cdots, v_n\}$ is a basis in $V$. Then, for…
Daniil
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If $A$ is full column rank, then $A^TA$ is always invertible

I need to prove that If $A$ is full column rank, then $A^TA$ is always invertible. I know when an $m \times n$ matrix is full column rank, then its columns are linearly independent. But nothing more to use to prove the above theorem. I'd…
Gigili
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Isomorphisms Between a Finite-Dimensional Vector Space and its Dual

So, we know that there is no "natural" isomorphism between a finite dimensional vector space $X$ and its dual $X^{\vee}$. However, given a basis for $X$ $(e_1, \dots, e_n)$ we can always construct a (dual) basis $(e^1, \dots, e^n)$ for $X^{\vee}$…
ItsNotObvious
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Is a linear transformation onto or one-to-one?

The definition of one-to-one was pretty straight forward. If the vectors are lin.indep the transformation would be one-to-one. But could it also be onto? The definition of onto was a little more abstract. Would a zero-row in reduced echelon form…
Algific
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How do you formally prove that rotation is a linear transformation?

The fact that rotation about an angle is a linear transformation is both important (for example, this is used to prove the sine/cosine angle addition formulas; see How can I understand and prove the "sum and difference formulas" in trigonometry?)…
guest
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Show that the direct sum of a kernel of a projection and its image create the originating vector space.

I got the following question as my homework. Given $V$ is a vector space with $P \in \operatorname{End} V$. $P \circ P = P$ ("P is idempotent"). Show that $V = \operatorname{Ker} P \oplus \operatorname{Im} P$. One $P$ I can imagine is a projection…
iblue
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What are the angle brackets in Linear Algebra?

In my linear algebra book, they have angle brackets around two different vectors, so it looks like this: $\langle\mathbf{u_2},\mathbf{v}_1\rangle$. They don't use angle brackets to define vectors, but use regular parenthesis instead. For the…
jfa
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Using the Left-Inverse to "Solve" an Impossible System of Equations

I was working with the following system of equations: $$\begin{split} \begin{bmatrix} 4 & 0\\ 0 & 5\\ 0 & 0\\ \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ \end{bmatrix} & = \begin{bmatrix} 1\\ 1\\ …
rish987
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Difference between epimorphism, isomorphism, endomorphism and automorphism (with examples)

Can somebody please explain me the difference between linear transformations such as epimorphism, isomorphism, endomorphism or automorphism? I would appreciate if somebody can explain the idea with examples or guide to some good source to clear the…
EngDR
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2 answers

How to diagonalize this matrix...

Can someone show me step-by-step how to diagonalize this matrix? I'm trying to teach myself differential equations + linear algebra, but I'm stumped on how to do this. I'd really appreciate if someone would take the time to do this with me!…
Josh
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Can the zero vector be an eigenvector for a matrix?

I was checking over my work on WolfRamAlpha, and it says one of my eigenvalues (this one with multiplicity 2), has an eigenvector of (0,0,0). How can the zero vector be an eigenvector?
user83039
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Finding the coordinate vector relative to a given basis

Given the basis $\beta = \{(1,-1,3),(-3,4,9),(2,-2,4)\}$ and $x = (8, -9, 6)$, I am to find the corresponding coordinate vector $[x]_\beta$. I claim that the coordinate vectors entries $x_1,x_2,x_3$ meet the following…
Andrew Thompson
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Dimension of the sum of two vector subspaces

$\dim(U_1+U_2) = \dim U_1 +\dim U_2 - \dim(U_1\cap U_2).$ I want to make sure that my intuition is correct. Suppose we have two planes $U_1,U_2$ though the origin in $\mathbb{R^3}$. Since the planes meet at the origin, they also intersect, which in…
St Vincent
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How to prove the distributive property of cross product

That is, how to prove the following identity: $$a \times (b+c) = a \times b + a \times c$$ where the $\times$ represents cross product of two vectors in 3-dimensional Euclidean space.
KunXian Xia
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