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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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True of False: Every 3-dimensional subspace of $ \Bbb R^{2 \times 2}$ contains at least one invertible matrix.

The true or false question states: "True of False: Every 3-dimensional subspace of $ \Bbb R^{2 \times 2}$ contains at least one invertible matrix." Here the $ \Bbb R^{2 \times 2}$ represents the space of all two by two matrices. It seems like this…
Squirrel-Power
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Properties of the A-transpose-A matrix

I believe that $A^TA$ is a key matrix structure because of its connection to variance-covariance matrices. In Professor Strang's linear algebra lectures, "A-transpose-A" - with this nomenclature, as opposed to $X'X$, for example - is the revolving…
Antoni Parellada
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Is reducing a matrix to row echelon form useful at all?

I have just started auditing Linear Algebra, and the first thing we learned was how to solve a system of linear equations by reducing it to row echelon form (or reduced row echelon form). I know how to solve a system of linear equations using…
Vivi
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Eigenvalues of Matrix vs Eigenvalues of Operator

I'm having some trouble reconciling the concept of eigenvalues of operators with eigenvalues of matrices: Say you have an $n\times n$ matrix $A$. It represents a linear operator $T:V\to V$ with respect to some basis $\{e_i\}$ in the background. Now…
user124910
  • 3,007
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1 answer

Linear algebra and arbitrary fields

The linear algebra course that I took was fairly consistent about assuming that the scalar field is either the reals or the complex numbers. The theory about linear maps, basis, their matrices, eigenvalues and eigenvectors, trace and determinant…
tkp
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Geometric argument that operators on $\mathbb{R}^3$ have an eigenvalue?

This question came up when trying to trying to find a $3\times3$ real matrix $A$ such that $Ax$ is nonzero for nonzero $x$ $Ax$ is orthogonal to $x$ for any $x$ in $\mathbb{R}^3$ We know such a matrix cannot exist because $A$ must have an…
Elliott
  • 4,124
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How exactly does the sign of the dot product determine the angle between two vectors?

I am told that $v\cdot w=0$ means that the angle between the vectors $v$ and $w$ is $90$ degrees. Then I am told that the sign of $v\cdot w$ (when it isn't equal to zero) determines whether the angle between vectors $v$ and $w$ is above or below…
Cherry_Developer
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When does $\det e^A=e^{\det A}?$

Which $2\times 2$ matrices satisfy the equation $$\det e^A=e^{\det A}?$$ I know that $\det e^A=e^{\operatorname{trace}A}$ so assuming $A$ is real we get $$\operatorname{trace}A=\det A.$$ Then, $$\det(A-\lambda…
asdasdasdasd
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Let $\lambda$ be an eigenvalue of $A$. Prove that $\lambda^{-1}$ is an eigenvalue of $A^{-1}$.

Let $\lambda$ be an eigenvalue of $A$. Prove that $\lambda^{-1}$ is an eigenvalue of $A^{-1}$. My approach: Suppose $\lambda$ is an eigenvalue of $A$. Then $Ax=\lambda x$ for some $x\neq 0$. Since $A$ is invertible, $Ax=\lambda x \implies…
user144809
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4 answers

Diagonalization of restrictions of a diagonalizable linear operator

I realized that I have some difficulties for prove this exercise. Let $T : V \rightarrow V$ be a linear operator on a finite dimensional vector space $V$ over a field $F$, and invariant subspaces $U,W \subset V$ such that $V = U \oplus W$. Show…
JimmyJP
  • 394
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3 answers

Linear operators on the functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that distribute over multiplication

Let $V$ denote the vector space of all functions $f:\mathbb{R}\rightarrow\mathbb{R}$. What are the linear operators $L:V\rightarrow V$ such that $L[fg]=L[f]L[g]$ for all $f,g\in V$? I made a bit of progress by considering the functions $$\chi_t(x) =…
user109360
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Proof of rank-nullity via the first isomorphism theorem

I was thinking about the proof of the rank-nullity theorem and I thought about proving it as follows. I just wondered whether this proof worked? Lemma. If $V$ is a finite-dimensional $F$-vector space and $U\leq V$, then $V/U$ is finite…
user168365
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Why is the condition enough for a matrix to be diagonalizable?

I've heard that for a matrix $A\in M_n(\mathbb{C})$, if $A^3=A$, then $A$ is diagonalizable. Does there happen to be a proof or reference as to why this is true? Out of curiosity, is it necessary that the entries be from $\mathbb{C}$? Would any…
Vika
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Example of a subset of $\mathbb{R}^2$ that is closed under vector addition, but not closed under scalar multiplication?

I've found several examples which are closed under scalar multiplication, but not vector addition, but I can't come up with one that is closed under vector addition, but not scalar multiplication.
Grid
  • 611
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2 answers

Spectrum of a linear operator on a vector space of countable dim

How would you prove that if V is a vector space over $\mathbb{C}$ of countably infinite dimension, and $T$ is a linear operator on V, then Spectrum($T$) is non-empty?
Rankeya
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