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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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Invariant subspace of cyclic space is cyclic

Let $V$ be a finite dimensional vector space and let $T:V\rightarrow V$ be a cyclic linear operator, that is, there exists $v \in V$ such that $\{v, Tv, T^2v, \dots\}$ generates $V$. Let $W\subset V$ be a $T$-invariant subspace, that is,…
user79594
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1 answer

Question about kernel of the restriction of a linear operator

I'm studying linear algebra and I am trying to answer a question I asked myself. Suppose $T:V\rightarrow V$ is a linear operator on a finite dimensional vector space $V$ over an algebraically closed field $K$. Let $\lambda_1, \dots, \lambda_n$ be…
user79594
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Plane determined by two lines

Show that the lines $x=-2+t,y=3+2t,z=4-t$ and $x=3-t,y=4-2t,z=t$ are parallel. Find the equation of the plane they determine. Here what is the meaning of "they determine"?
Raisul RONI
  • 71
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Is there any relation between the eigenvalues of possibly non-Hermitian matrix A and those of exp(A)?

Is there any relation between the eigenvalues of possibly non-Hermitian matrix A and those of exp(A)? For hermitian matrices, they are just exponentials of the corresponding values. But in general, any relation? Thanks.
Qiang Li
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Uniqueness of orthogonal projector

Let $S \subseteq \mathbb{R}^n$ be a subspace of $\mathbb{R}^n$, and let $P_1$, $P_2$ be arbitrary orthogonal projectors onto $S$. How can I prove that the orthogonal projector onto $S$ is unique, i.e. that $P_1 = P_2$?
mahya
  • 69
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Complexification of a real Algebra

Let $\mathbb R$ be the field of real numbers and $\mathbb C$ be the field of complex numbers. Consider the complexification of the real matrix algebra $M_n(\mathbb R)$ that is $\mathbb C\otimes_{\mathbb R}M_n(\mathbb R)$. It is known that $$\mathbb…
zacarias
  • 3,158
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Orthogonal eigenvectors in symmetrical matrices with repeated eigenvalues and diagonalization

Symmetrical matrices have orthogonal eigenvectors. However, there is the special case when eigenvalues are repeated. The ultimate scenario is that of the identity matrix. Professor Strang mentions here that "if an eigenvalue is repeated, then there…
Antoni Parellada
  • 8,889
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how to express the result of triple cross product of two vectors in spherical coordinates (unit vector)

I have a system of differential equations like below: (real problem is more complex, here is just an…
Enlong Liu
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Complexification of a self-adjoint operator is self-adjoint on the complexified space

In "Linear algebra done right" 9.b.4 : Suppose $V$ is a real inner product space and $T \in \mathcal{L}(V)$ is self-adjoint. Show that $T_\mathbb{C}$ is a self-adjoint operator on the inner product space $V_\mathbb{C}$. My way to do it is as…
Sean Zhou
  • 309
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If two vector spaces $V$ and $W$ are isomorphic and $V$ is F-D then $W$ is F-D. Furthermore, $\text{dim} V = \text{dim} W$.

Is the following statement true? Conjecture. If two vector spaces $V$ and $W$ are isomorphic and $V$ is finite dimensional (F-D) then $W$ is finite dimensional. Furthermore, $\text{dim} V = \text{dim} W$. and if YES, then how it can be proved? I…
Hosein Rahnama
  • 14,882
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Eigenvalue of a polynomial evaluated in a operator

Suppose $T:V\to V$, $p\in \mathcal{P}(\mathbb{C})$ (polynomials with complex coefficients), and $a\in \mathbb{C}$. Prove that $a$ is an eigenvalue of $p(T)$ if and only if $a=p(\lambda)$ for some eigenvalue $\lambda$ of $T$. I can prove: if…
Hiperion
  • 1,763
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An operator $T:\mathbb{R}^4\to \mathbb{R}^4$ such that $T$ has no (real) eigenvalues.

Give an example of an operator $T:\mathbb{R}^4\to \mathbb{R}^4$ such that $T$ has no (real) eigenvalues. How can I find this operator? Thanks for your help.
Hiperion
  • 1,763
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vector space without verification of the axioms

I'm trying to show that the functions $c_1 + c_2 \sin^2 x + c_3 \cos^2 x$ forms a vector space. And I will need to find a basis of it, and its dimension. Is there a way how to do this without verifying the 8 axioms for a vector space, and if we let…
mary
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Typo in Hoffman and Kunze's linear algebra book

I'm reading Hoffman and Kunze's linear algebra book and on page 183 they made this claim: I think it must be "when its determinant is $0$" instead of "when its determinant is different from $0$". Am I right?
user42912
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I am confused by the statement "the null space of A is a nontrivial"

Correct me if I'm wrong but if a null space of a matrix A is nontrivial would it be correct to say that it is the opposite of the list of points in the Invertible Matrix Theorem? A is an invertible matrix A is row equivalent to the identity…
MasterExploder
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