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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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If $V_0$ is the subspace of matrices of the form $C=AB-BA$ for some $A,B$ in a vector space $V$ then $V_0=\{A\in V|\operatorname{Trace} (A)=0\}$

If $V_0$ is the subspace consisting of matrices of the form $C=AB-BA$ for some $A,B$ in a vector space $V$ then $V_0=\{A\in V|\operatorname{Trace}(A)=0\}$. The problem above is one of the past qualifying exam problems. I can prove that …
Yeonjoo Yoo
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Intuition behind speciality of symmetric matrices

What is the geometric intuition behind the fact that only matrices that are similar to a symmetric matrix are diagonizable? So e.g. why is it important that the multiplier of the the first component of the last basis vector be the same as the…
Nesa
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How can I visualize independent and dependent set of vectors?

Can someone help me visualize those concepts? It will also help me understand it better. Thanks :)
LiziPizi
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Why is this set of polynomials linearly dependent?

$$1 + 2t+ t^2, 3-9t^2,1 + 4t + 5t^2$$ (A) Linearly dependent or (B) Linearly independent The answer is A from the answer key. This is a test review. I don't see that either polynomial is a scalar multiple of any of the other…
asdf
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Is the dual space of a subspace of a vector space simply its orthogonal complement?

Is the dual space of a subspace of a vector space simply its orthogonal complement? My professor seems to use the terms interchangeably, but from Wikipedia they seem to be quite distinct concepts (like dual spaces are much more complicated and…
badatmath
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What kind of vector spaces have exactly one basis?

Here is the question as an exercise in the book Linear Algebra Done Right, Chapter 2 Find all vector spaces that have exactly one basis.
When
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A question about an $n$-dimensional subspace of $\mathbb{F}^{S}$.

I am self-studying Hoffman and Kunze's book Linear Algebra. This is Exercise 3.6.3(Linear Transformation-The Double Dual) from page 111. Let $S$ be a set, $\mathbb{F}$ a field, and $V(S,\mathbb{F})$ the space of all functions from $S$ into…
user23505
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Finding a basis for subspace of polynomials

Let $V=\mathscr{P}_{3}$ be the vector space of polynomials of degree 3. Let W be the subspace of polynomials p(x) such that p(0)= 0 and p(1)= 0. Find a basis for W. Extend the basis to a basis of V. Here is what I've done so far. $$p(x) = ax^3 +…
CloudN9ne
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$\det(I+\epsilon V)=1+\operatorname{trace}(V)\epsilon+O(\epsilon^2)$

How to show that $$\det(I+\epsilon V)=1+\operatorname{trace}(V)\epsilon+O(\epsilon^2)$$ for any $n\times n$ real matrix $V$? This is used a lot in the theory Lie groups, but I never saw a proof of it.
Peter
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If every vector is an eigenvector, the operator must be a scalar multiple of the identity operator?

I am posed with the following question: Suppose that $T\in \mathcal{L}(V)$, where $V$ is a finite-dimensional vector space, is such that every vector in $V$ is an eigenvector of $T$. Prove that T is a scalar multiple of the identity function. My…
Arturo don Juan
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Why do complex eigenvalues correspond to a rotation of the vector?

We have a linear transformation $T: \Bbb R^m \to \Bbb R^n$ defined by $T(x)=Ax$ for $x \in \Bbb R^m$ and $A \in M_{n \times m}(\Bbb R)$. I understand why real-valued eigenvalues of $A$ correspond to scaling the length of the associated…
got it--thanks
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Minimal polynomials and cyclic subspaces

I'm trying to make my way through two problems in Curtis's Linear Algebra, chapter 25. One of the two problems is this one, #5: Prove that $V$ is cyclic relative to a linear transformation $T \in \mathcal{L}(V)$ if and only if the minimal…
dmk
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What is the motivation/application of dual spaces and transposes?

I've always been baffled as to where transposes come from. I found this question, but the answer isn't satisfying to me - the idea seems to be "dual spaces are important, and you can define transposes using those". This leaves two questions: Why…
Jack M
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Prove that : $a^n+b^n+c^n=x^n+y^n+z^n$; $\forall n\in \mathbb{N}$

$a;b;c;x;y;z \in \mathbb{R}$ such that : \begin{matrix} a+b+c=x+y+z & \\ a^2+b^2+c^2=x^2+y^2+z^2 & \\ a^3+b^3+c^3=x^3+y^3+z^3 & \end{matrix} Prove that : $a^n+b^n+c^n=x^n+y^n+z^n$; $\forall n\in \mathbb{N}$ P/s : I don't have any ideas about…
Lê Tấn Khang
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Connection between linear independence, non-/trivial and x solutions

I am having a hard time remembering which goes hand in hand with what. The math questions I get always include words like trivial etc. 1 solution no solution infinite amount of solutions And then we have the two types of set of vectors linearly…
Karl Morrison
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