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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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Non linear transformation satisfying $T(x+y)=T(x)+T(y)$

Given V a vector space with vectors and scalars $\mathbb{C}$, does there exists a non linear transformation $T:V\rightarrow V$ such that $T(x+y)=T(x)+T(y)$ for all $x,y\in V$? I think such a transformation will be 'like' one that satisfies Cauchy's…
Ivan Wangsa
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How to find sides of parallelogram given centroid and two vertices?

I've been given a parallelogram with two of its vertices, $A$ and $B$, being equal to (-5, -8, 3) and (4, 7, -5) respectively, and a centroid $S$ at (-10, 4, 6). How do I go around finding remaining coordinates of points $D$ and $C$?
J.Doe
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Resultant of two polynomials

Let $Res(f,g)$ be a resultant of two polynomials $f(x)$ and $g(x)$. Is it true that resultant does not change under a linear change of coordinates $x\mapsto x+\alpha$? Thanks a lot!
Aspirin
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When is the union of a family of subspaces of a vector space also a subspace?

It is not difficult to prove that the union of a chain (or, more generally, a directed family) of subspaces of a vector space $V$ is a subspace of $V$. Given a family $\mathcal{F}$ of subspaces of a vector space $V$ such that the union of…
user355272
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Linear algebra proof with linear operator

Let $V$ be a finite dimensional vector space, $T \in L(V)$ a linear map whose matrix $M(T)$ is the same for every basis of $V$. Show that $T$ is a scalar multiple of the identity map $I$. I know it has to do with something about the vectors being…
fireshock
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Intersection and Span

Assume $S_{1}$ and $S_{2}$ are subsets of a vector space V. It has already been proved that span $(S1 \cap S2)$ $\subseteq$ span $(S_{1}) \cap$ span $(S_{2})$ There seem to be many cases where span $(S1 \cap S2)$ $=$ span $(S_{1}) \cap$ span…
mathnoob
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Four Fundamental subspaces

Given an $m \times n$ matrix, where $m$ is the number of rows and $n$ is the number of columns. Four Fundamental Subspaces The row space is $C(A^t)$, a subspace of $\mathbb{R}^n$. The column space is $C(A)$, a subspace of $\mathbb{R}^m$. The…
aceminer
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Counting $T$-invariant subspaces over finite field

I have some explicit questions regarding the following problems. Additional critiques/suggestions are more than welcome. Let $F$ be a finite field with $p$ elements, let $V$ be a $3$-dimensional vector space over $F$ and let $T\colon V\to V$ be a…
user346096
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Basic Understanding of Linear Combinations Geometrically

I am currently working through MIT's Introduction to Linear Algebra by Gilbert Strang, with no previous matrix experience. In the first lecture, we are given the following linear equations: $$2x - y = 0\\ -x + 2y = 3$$ The solution to the system…
Andrew Doherty
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Find all real numbers $t$ such that the quadratic form $f$ is positive definite.

Where $$f(x_1,x_2,x_3)=2x_1^2+x_2^2+3x_3^2+2tx_1x_2+2x_1x_3$$. This is a problem in my Matrix Analysis homework. Below is my effort. Let $x=(x_1,x_2,x_3)^T$, then we have $$f=x^*Sx$$, in which…
xzhu
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Does there exist a normal matrix which is not diagonal, nor Hermitian nor Unitary nor Skew Hermitian?

Well, we know that if a matrix is diagonal, Hermitian, Unitary or Skew-hermitian, then the matrix is Normal. But is the converse true? If not, can anyone give an example?
user198504
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Relationship between properties of linear transformations algebraically and visually

I learned from 3Blue1Brown's Linear Algebra videos that a 2-D transformation is linear if it follows these rules: lines remain lines without getting curved the origin remains fixed in place grid lines remain parallel and evenly spaced I'm now…
Neil Aronson
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Compute the equivalence classes

Define an equivalence relation on $\mathbb{R}^2$ by $\textbf{x}\sim\textbf{y}$ iff $\exists A\in GL_2(\mathbb{R})$ such that $A\mathbf{x}=\mathbf{y}$. Compute the equivalence classes of this equivalence relation. My attempt: Let…
Siddhartha
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Can orthogonal matrix be complex?

A matrix $Q$ is orthogonal if $Q^TQ=I$. My question is can $Q$ be complex? Can anyone help show an example, or provide a proof that $Q$ has to be real? Thanks.
Ralph B.
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Prove that $V=U \bigoplus W \approx U \times W$

Let $V=U \bigoplus W$, $V \approx U \times W$. Note that $U,W$, are finite dimensional subspaces of the vector space V, and also that $U \bigoplus W$ means $V=U+W$ and $U \cap W = \{0\}$ I'm really not sure how to go about this, because it doesn't…
tk2
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