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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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How to verify whether R^2 is a subspace of the complex vector space C^2?

It's an exercise of the book Linear Algebra Done Right. I'm not clear about how to prove these problems, would you please offer me some suggestion about how to improve this kind of ability, thanks a lot.
When
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A question on "Linear Algebra" by Kenneth Hoffman

I'm reading "Linear Algebra" by Kenneth Hoffman and Ray Kunze. I'm now lost at $\S$6.4 Theorem 6: the proof looks OK, but when I pick an example, somehow it does not tally. Please find below the theorem and proof, and my example in…
athos
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Show that $V = \mbox{ker}(f) \oplus \mbox{im}(f)$ for a linear map with $f \circ f = f$

Question: Let $V$ be a $K$-Vectorspace and $f: V \rightarrow V$ be linear. It holds that $f \circ f = f$. Show that $V = \mbox{ker}(f) \oplus \mbox{im}(f)$. My attempt: So i guess that the $\oplus$ denotes a direct sum which means i have to show…
ghshtalt
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Eigenvectors of a complex matrix

Given the following matrix $\begin{pmatrix} 0 & 1-i & 0\\ 1+i & 0 &1-i\\ 0& 1+i &0\\ \end{pmatrix}$ I have found the Eigenvalues $0, 2,-2$. But I have no idea how to calculate the corresponding Eigenvectors and I failed with Gaussian method. What…
user221151
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How can an engineer make use of commutative diagram?

I am in engineering and I have being awed by commutative diagram my entire life. But I do not see the purpose in knowing commutative diagram as arises in linear algebra? Is there something in engineering that can be describe by commutative diagram?…
Olórin
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Orthogonal complement of subspace of continuous functions and odd in the interval $[-1,1]$.

The follow question was found on the Hoffman's book. Let $V$ be the real inner product space consisting of the space of real-valued continuous functions on the interval, $-1\leq t \leq 1$, with the inner product $(f|g)=\displaystyle \int_{-1}^{1}…
Pitágoras
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Is there a converse for the following Theorem? Equivalent systems of linear equations have exactly the same solutions

On page $5$ of Linear Algebra from Hoffman and Kunze's book they prove the following Theorem: Theorem 1. Equivalent systems of linear equations have exactly the same solutions I will add the definition of equivalent system of linear…
user23505
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To show that $\operatorname{Rank}(\mathbf{A}-\mathbf{I})=\operatorname{Nullity}(\mathbf{A})$

Problem is: Let $\mathbf{A}$ be $n\times n$ matrix with real entries such that $\mathbf{A}^{2} = \mathbf{A}$. If $\mathbf{I}$ denotes the identity matrix, then how do I prove the result:…
godonichia
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On the importance of order for bases in finite dimensional vector spaces

I am reading Tapp's Introduction to Matrix Groups for Undergraduates and he writes: Let $V$ be an $n$-dimensional (left) vector space over $\mathbb K$. Then $V$ is isomorphic to $\mathbb K^n$. In fact, there are many isomorphisms from $V$ to…
learner
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What is the difference between matrix theory and linear algebra?

I have lifted this from Mathoverflow since it belongs here. Hi, Currently, I'm taking matrix theory, and our textbook is Strang's Linear Algebra. Besides matrix theory, which all engineers must take, there exists linear algebra I and II for math…
user218
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Let V be a vector space. If every subspace of V is T-invariant, prove that there exist a scalar multiple c such that T=c1v

I wrote a proof for the above question, but I am not sure whether it is right or not since I assumed linear independence. Here's the proof: Let $u$,$v$ be linearly independent vectors in $V$. $span(u)$, $span(v)$, $span(u+v)$ are all…
javabeginner_0101
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A projection satisfying $\| Px \| \leq \|x\|$ for all $x$ is an orthogonal projection

How to prove that if $V$ is a finite dimensional inner product space and $W$ a subspace of $V$, if $P$ is projection map ($P^2=P$) having $W$ as its range and is such that $\|Px\| \leq \|x\|$ for all $x \in V$, then $P$ is orthogonal projection of…
Abcd J
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Center of general linear group

Given a (not necessarily finite dimensional) vector space $V$ prove that the center of $\operatorname{GL}(V)$ is the set of all scalar transformations (i.e all transformations of the form $a\operatorname{Id}$)? I know how to prove this for general…
Dopeman
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If A is positive definite, can we prove that x>0 and Ax>0 always have a feasible solution?

I'm new here and couldn't find a similar question, so pardon me if it's already asked elsewhere. The question is literally simple: Suppose A is a positive definite matrix , could it be generally proved that the set $S=\left \{ x\in R^{n} |x>0,Ax>0…
Iman E
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Unique solution comes out to be trace

Let $f: \mathcal{M}_n(F)\to F$ be a linear functional satisfying $f(\mathbf{AB})=f(\mathbf{BA})\;\forall \mathbf{A,B}\in \mathcal{M}_n(F)$. Also it satisfies $f(\mathbf{I})=n$. Prove that $f(\mathbf{A})=\operatorname{tr}(\mathbf{A})$. Can someone…
shadow10
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