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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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Every $v \in V - \{ 0 \}$ is cyclic iff the characteristic polynomial of $T : V \to V$ is irreducible over $F$

Let $V\neq \{0\}$ be a vector space over $F$, and $T$ a linear operator on $V$. Prove that every $0\neq v \in V$ is a cyclic vector if and only if the characteristic polynomial of $T$ is irreducible over $F$. I can't seem to get anywhere on either…
izikgo
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Angle preserving linear maps

In Spivak's Calculus On Manifolds, in part (c) of question 1-8, he asks the following question: What are all angle preserving $T:\mathbf{R}^n \to \mathbf{R}^n$? I already showed that if $T$ is diagonalizable with a basis $\{x_1,\ldots,x_n\}$ where…
nigel
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If three corners of a parallelogram are known solve for the 3 possible 4th corners.

An example would be three corners being the points: (1,1), (4,2) and (1,3). I understand the specific solution for this example: (4,4), (4,0) or (-2,2). Which I reasoned when i drew it out. The example came from a linear algebra textbook and i'm…
Drew Verlee
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A problem about a rank's inequality of a complex matrix

Let $X,Y \in M_{n\times n}(\mathbb{C})$ such that $X^{2}=Y^{2}=I_{n}$. Prove that $$\operatorname{rank}((X+I_n)(Y-I_n))+\operatorname{rank}((X-I_n)(Y+I_n))=\operatorname{rank}(X-Y)$$ My approach: Using the well-known inequality:…
user798113
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does a matrix like this exist?

Question: Does a matrix $A \in M_{3 \times 3}(F)$ exist s.t. $A^4= \begin{bmatrix} 0&0&1\\0&0&0\\0&0&0\end{bmatrix}$ What I thought: I think it doesn't. How do you start a proof of such a thing (prefer hints at first). Thx
jreing
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Dimensions of vector subspaces in a direct sum are additive

$V = U_1\oplus U_2~\oplus~...~ \oplus~ U_n~(\dim V < ∞)$ $\implies \dim V = \dim U_1 + \dim U_2 + ... + \dim U_n.$ [Using the result if $B_i$ is a basis of $U_i$ then $\cup_{i=1}^n B_i$ is a basis of $V$] Then it suffices to show $U_i\cap…
Sriti
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One dimensional subspaces of a 3 dimensional vector space over $\mathbb Z/3\mathbb Z$, multiple choice

Let $V$ be a 3-dimentional vector space over the field $F_3=\Bbb Z/3 \Bbb Z$ of $3$ elements.the number of distinct 1 dimentional subspaces of $V$ is $13$ $26$ $9$ $15$
kinkar
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Is it possible that $\ker(T) = \operatorname{im}(T)$ for some linear transformation $T:V \to V$?

Help would be very appreciated. It it possible that $\ker(T) = \operatorname{im}(T)$ for some linear transformation $T:V \to V$?
matan
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A problem on subspaces

I was studying for some quals and I remember running into this problem last year and I couldn't get anywhere with it. Even now I'm kind of stumped. I was wondering if you guys had any ideas. Here's the problem: Let $ V $ be a vector space and let $…
poopstraw
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Orthonormal columns implies orthonormal rows

I find it non-intuitive if I impose that all of a square matrix's columns are normalized and mutually orthogonal, then all its rows are also normalized and mutually orthogonal. Any intuitive explanation for this? Also if I relax the conditions to be…
Sam
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Finding squared norm of vector

Suppose you have a $2\times 1$ column vector $x=[7,2]^{T}$. How would you find $||x||^{2}$? Would it be $7^{2} + 2^{2}$? Is this equivalent to the distance from the origin?
phil12
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Number of bases of an n-dimensional vector space over q-element field.

If I have an n-dimensional vector space over a field with q elements, how can I find the number of bases of this vector space?
käyrätorvi
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How do i prove that $\det(tI-A)$ is a polynomial?

In wikipedia, it's said "$\det(tI-A)$ can be explicitly evaluated using exterior algebra", but i have not learned exterior algebra yet and i just want to know whether it is polynomial, not how it looks like. How do i prove that $\det(tI-A)$ is a…
Jj-
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The annihilator of an intersection is the sum of annihilators

Given a subset $X$ of a vector space $V$, let $X^\circ$ be the annihilator of $X$, that is $X^\circ = \{y\in V^* | \; y(x)=0, \;\forall\; x\in X\}$, where $V^*$ is the dual space of $V$. Question: If $M$ and $N$ are subspaces of an $n$-dimensional…
Pedro
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This question appeared in a mate's child's homework and it's broken my brain.

I could use gaussian elimination if I make some assumptions or does any one have another suggestion?
chris loughnane
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