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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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Understanding how to find a basis for the row space/column space of some matrix A.

I just need some verification on finding the basis for column spaces and row spaces. If I'm given a matrix A and asked to find a basis for the row space, is the following method correct? -Reduce to row echelon form. The rows with leading 1's will be…
briteId
  • 353
6
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1 answer

reflection representation of isometry

I am reading the book Naive Lie Theory It proves that any isometry of $R^n$ that fixed the origin O is the product of at most n reflections in hyperplanes through O. The proof is elementary and by induction. However, I cannot understand the…
noot
  • 757
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Every matrix can be changed to a symmetric matrix through elementary column operations

The following question is given in a section 2 lecture of linear algebra. The first section is about polynomial, so the lectures just started to talk about determinants and matrices. Let $A$ be an $n\times n$ matrix over a number field $F$. Then…
Dan Sims
  • 515
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If every eigenvector of $T$ is also an eigenvector of $T^{*}$ then $T$ is a normal operator

Let $V$ be a finite-dimensional inner product space over $\mathbb{C}$ and $T: V \to V$ a linear transformation. Show that if every eigenvector of $T$ is also an eigenvector of $T^{*}$ then $T$ is a normal. We need to show that $\forall v \in V,\…
daniel.jackson
  • 2,361
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2 answers

Prove that the integral operator has no eigenvalues

Let $V$ be the vector space of all real valued continuous functions. Prove that the linear operator $\displaystyle\int_{0}^{x}f(t)dt$ has no eigenvalues.
Grobber
  • 3,248
6
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2 answers

Number of solutions of a linear equation $AX=B$

I have a question and a proposed solution - Please tell me if I'm correct. Problem: Prove that if $A$ and $B$ are real matrices and the system of equations $AX=B$ has more than one solution, then it has infinitely many. Solution: Assume that the…
user85362
6
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1 answer

Maximum number of linearly independent non-commuting matrices

Let $S$ be a set of non-commuting, linearly independent $d \times d$ positive definite matrices (i.e., for any $A \neq B$, $[A, B] = AB - BA \neq 0$). Is there any upper bound for the number of elements the set $S$ contains? (It is clear that it…
Hsyn
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Existence of a matrix, characteristic polynomial and minimal polynomial

Let $k$ be a field and let $p(x),q(x)$ be elements of $k[x]$. If $q(x)$ divides $p(x)$ and if every root of $p(x)$ is a root of $q(x)$ prove there exists a matrix $A$ having minimal polynomial equal $q(x)$ and characteristic polynomial equal…
user10
  • 5,688
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3 answers

Is this a linear transformation?

Let $T$ be a transformation from $P_2$ to $P_2$ (where $P_2$ is the space of all polynomials with degree less than or equal to $2$) $$T(f(t)) = f''(t)f(t)$$ I'm tempted to say that this is not a linear transformation because $$T(f(t) + g(t)) =…
Andy
  • 95
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$1, e^{ix}, e^{-ix}$ are linearly independent

Consider the space of all functions $f: \mathbb{R}\longrightarrow \mathbb{C}$. Prove that $\{1, e^{ix}, e^{-ix}\}$ are linearly independent vectors.
Grobber
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2 answers

How does one combine proportionality?

this is something that often comes up in both Physics and Mathematics, in my A Levels. Here is the crux of the problem. So, you have something like this : $A \propto B$ which means that $A = kB \tag{1}$ Fine, then you get something like : $A \propto…
Nafiul Islam
  • 579
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Bijective map preserving inner products is linear

The question comes from Kaplansky's book Linear Algebra and Geometry on page 96 exercise 2 Let $V$ be a non-singular inner product space of characteristic $\neq2$. Let $T$ be a one-to-one map of $V$ onto itself, sending $0$ to $0$ and satisfying…
ADR
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How to determine the rank and determinant of $A$?

let $A$ be $$A_{a} = \begin{pmatrix} a & 1 & 1 & 1 \\ 1 & a & 1 & 1\\ 1 & 1 & a & 1\\ 1 & 1 & 1 & a \end{pmatrix}$$ How can I calculate the rank of $A$ by the Gauss' methode and $\det A$?
justme
  • 61
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5 answers

If $\dim V=v$ and $\dim(\ker T)=n$, prove that $T$ has at most $v-n+1$ distinct eigenvalues

Let $T:V\to V$ be a linear operator. If $\dim V=v$ and $\dim(\ker T)=n$, prove that $T$ has at most $v-n+1$ distinct eigenvalues. I have been working on this proof for a few days and I am not sure what direction to really go with it? I feel like…
Joe
  • 153
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5 answers

proof about commutative operators and T-cyclic vectors

Let $V$ be a finite dimensional vector space over $F$. Let $T:V \to V$ be a linear operator. Prove that if every linear operator $U$ which commutes with $T$ is a polynomial of $T$, than $T$ has a $T$-cyclic vector. I don't really know where to…
izikgo
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