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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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Prove that if $A$ is invertible matrix then $AA^T$ and $A^T A$ are also invertible.

I want to know how to prove this question: Prove that if $A$ is invertible matrix then $AA^T$ and $A^T A$ are also invertible. My attempt: Since $A$ is invertible we have that $AA^{-1} = I$ and if we denote $B = A^{-1}$, we have that $AB=I$ so…
diimension
  • 3,410
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Linear Algebra Proof $AB=0 \Longrightarrow \det(A)=0$

Two squared matrices $A$ and $B$, with $B\neq0$, give $AB=0$. Prove that $\det(A)=0$. After trying with some examples, I believe that $A$ needs to have lines that are equal or can be made equal by scalar multiplication, B needs to have columns that…
Anna Sarah
  • 181
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Prove that vectors in Euclidean space are linearly independent

Could you help me with an idea of solving the following problem? I think that proof involves the positive definiteness of Gram matrix, but I don't know how. Consider a system of vectors $e_1, e_2, ..., e_n, e_{n+1}$ in some Euclidean space such that…
Dan
  • 353
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Finding solutions for a linear system of equations

Got the following problem where I can't find a way to solve: Knowing $\begin{pmatrix}5\\ 3\\ 6\end{pmatrix}$ is the unique solution for the system $Ax=\begin{pmatrix}2\\1\\1\end{pmatrix}$, with $A \in \mathbb{R}^{3\times3}$ and $B=\begin{pmatrix} 1…
Lucas
  • 75
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4 answers

Kernel of Linear Functionals

Problem: Prove that for all non zero linear functionials $f:M\to\mathbb{K}$ where $M$ is a vector space over field $\mathbb{K}$, subspace $(f^{-1}(0))$ is of co-dimension one. Could someone solve this for me?
Parakee
  • 3,304
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Is there a general rule for how to write high order polynomials in matrix form?

Is there a general rule for how to write high order polynomials in matrix form? For example a linear combination of parameters: $$a_1 x_1+a_2 x_2+a_3 x_3 + \cdots+ a_n x_n$$ Can be written as $$\sum^n_{i=1} a_i x_i = \vec{a}^T\vec{x} $$ Second…
user2350366
  • 513
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Prove that there is a vector $v\in \mathbb{R}^k$ such that $u \cdot v =0$

Let $u \in \mathbb{R}^k$ be a vector with one component positive, one component negative, and the remaining $k-2$ can have at most one component that is equal to zero. Then is there a vector $v \in \mathbb{R}^k$ such that all its components are…
satokun
  • 677
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Show that $(1,1,1)$, $(a,b,c)$, $(a^2, b^2, c^2)$ are linearly indepdenent for distinct $a,b,c$

Show that $(1,1,1)$, $(a,b,c)$, $(a^2, b^2, c^2)$ are linearly indepdenent, where $a,b,$ and $c$ are distinct real numbers. I will show my attempt and then state where I get stuck. Suppose $c_1(1,1,1) + c_2(a,b,c) + c_3(a^2,b^2,c^2) = 0$ This leads…
IntegrateThis
  • 3,778
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How to find the linear transformation associated with a given matrix?

Good day, I have a little doubt: It is well known that given two bases (or even one if we consider the canonical basis) of a vector space, every linear transformation $T:V \rightarrow W$ can be represented as a matrix, but since this is an…
Hendrik Matamoros
  • 659
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Dimension of Range and Null Space of Composition of Two Linear Maps

Question Suppose $U$, $V$ and $W$ are finite-dimensional vector spaces. Let $\mathcal{L}(U,V)$ and $\mathcal{L}(V,W)$ be the vector spaces of all linear maps from $U$ into $V$ and from $V$ into $W$, respectively. Suppose $S \in \mathcal{L}(V,W)$ and…
Hosein Rahnama
  • 14,882
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Properties of $f^{\star}$

Doing some linear algebra exercises I found that: Given $f \in \mathcal{End}(V)$ we define $f^\star$ an endomorphism such that, given $\phi$ a positive scalar product: $\phi(f(x),y)=\phi(x,f^\star(y))$. Let $V$ a finite dimensional vector space,…
JCF
  • 653
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1-dimensional solution space of homogeneous system Ax=0?

Given is an almost-square matrix $A$ with $n$ columns and $n-1$ rows with maximum rank. The solutions of the homogeneous system $Ax = 0$ form a 1-dimensional subspace of $\mathbb{R}^n$. I've discovered the following which I believe to be true but I…
user6216
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2 answers

Solution(s) to dot product of vectors

I have some questions about the uniqueness of matrices when post- and pre-multiplied with vectors (inner product). Say we have two vectors $\vec{a}$ and $\vec{b}$, whose inner product is a scalar, known to satisfy the following equation involving…
Stuart Barth
  • 301
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Finding range of a linear transformation

Define $T: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ by $T(x,y,z) = (2y + z, x-z)$. Find $\mbox{ker}(T)$ and $\mbox{range}(T)$ I could find the kernel easy enough, and ended up getting $\{(-2x, x, -2x) : x \in \mathbb{R}\}$ but I don't really know how…
user1520427
  • 625
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Algebraic multiplicity = geometric multiplicity?

I was wondering if algebraic multiplicity was equal to the geometric multiplicity. If the matrix (of size $n\times n$) is diagonalisable, i.e. the characteristic polynomial is of the form $$p(x)=(x-\lambda_1)^{m_1}\cdot ...\cdot…
user349449
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