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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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Eigenvalues of the linear system $XA +A^T X = 0$

I am following a proof of the next theorem, rephrased from Theorem 8.5 of the book "Introduction to the theory of differential inclusions" by Georgi V. Smirnov. Let $\{\lambda_1,\dots,\lambda_n\}$ be eigenvalues of a constant matrix $A \in…
flyingwith
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What really is the difference between $\mathbf{v}=(2,3) $ and $\mathbf{u}=(2,3,0)$?

I used to believe that there is no difference between $\mathbf{v}=(2,3)$ which is a vector lying in $xy$ plane and $\mathbf{u}=(2,3,0)$ which is a three dimensional vector but still lying in $xy$ plane. So, for me both the vectors $\mathbf{v}$ and …
Vidyanshu Mishra
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Let $U$ and $V$ be vector spaces, and $T_1$ and $T_2$ be linear maps from $U$ to $V$ and $V$ to $U$ respectively, and are onto maps.

Let $U$ and $V$ be vector spaces, and $T_1$ and $T_2$ be linear maps from $U$ to $V$ and $V$ to $U$ respectively, and are onto maps. Are $U$ and $V$ isomorphic? If both space are finite dimensional then they are isomorphic. But in other cases? I…
Eklavya
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Two questions with respect to the determinants

I have got a proof of $det(AB)$=$(detA)(detB)$ in my book. It goes as follows (for invertible A): we know that rref[A|AB]=[$I_{n}$|B] We also know that det(A)=$(-1)^{s}k_{1}k_{2}...k_{r}$ where s is the number of row swaps needed to get to the…
LinAlgStudent
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Finding eigenvalues shortcut

I am being asked to find the eigenvalues for this matrix. It mentions that some tricks can be used instead of having to use $det(A-\lambda I)$. I understand how to do it that way, but what is a shortcut I can use for this matrix? Thanks. Solution is…
cisco
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To show a function is bilinear symmetric non degenerate form

Let $V$ be vector space of set of $n×n$ matrices over $R$. Define $\langle A,B \rangle = \mathrm{trace}(AB)$, $A$, $B$ in $V$. show that $\langle \ \ ,\ \rangle$ is a non degenerate symmetric bilinear form. Now succeeded in showing that the…
Kavita
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Triangle equality implies vector dependence.

I am trying to prove this statement: Show that if $x$ and $y$ are two vectors in an inner product space such that $||x+y||=||x||+||y||$, then $x$ and $y$ are linearly dependent. Squaring the equality I get $$\langle x+y,x+y\rangle=\langle…
Jimmy R
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Map not preserving vector addition but preserving scalar multiplication

The question Map closed under addition but not multiplication asks for a map between two vector spaces where vector addition is preserved but also where scalar multiplication is not preserved. A student of mine switched this, and I have not found an…
Daniel Kiteck
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This algorithm will find a basis for the span of some vectors. How/why does it work?

Say I want to find a basis for $span((1,2,5),(2,4,10),(-3,-5,-13),(2,1,4),(-4,-6,-16))$. Google tells me that to get the answer, I'm supposed to write down the vectors as columns of a matrix: $$ \begin{pmatrix} 1 & 2 & -3 & 2 & -4…
Cassiterite
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How to find a basis for this sub-space?

I am given a subspace of all polynomials $f(t)$ in $\mathbf{P}_2$ such that $f(1)=0$. I know that a basis for this space is $1-t$, $1-t^{2}$, and when I look at it, it makes perfect sense as to why. I was just wondering what is a systematic way of…
user7016
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Sparse basis orthogonal to the ones vector

Let $v\in \mathbb{R}^{n}$ be the vector of ones, $v=(1,1,1,\cdots,1).$ I need an orthogonal basis for the orthogonal complement $v^{\perp}$, the space of all vectors orthogonal to $v$. Of course, one can solve for such a basis using Gram-Schmidt,…
user7530
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Number of vectors in a set & span of a set

I needed clarification on a linear algebra question that I had: Given the matrices $v_1 = \begin{bmatrix} 1 \\ 1 \\ 1 \\ \end{bmatrix}, $ $v_2 = \begin{bmatrix} 1 \\ -1 \\ 1 \\ \end{bmatrix}$ and $v_3 = \begin{bmatrix} …
user400359
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Explain why a set of mutually orthogonal non-zero vectors is linearly independent given a clause

"Given $\vec{u}_1,\ldots ,\vec{u}_n$ mutually orthogonal non-zero vectors, explain why for $\vec{v}=c_1\vec{u}_1+\ldots +c_n\vec{u}_n$ $c_k=\frac{\vec{v} \cdot \vec{u}_k}{\vec{u}_k \cdot \vec{u}_k}$" This I explained by dotting both sides with…
LinAlgStudent
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Showing that linear subset is not a subspace of the Vector space $V$

I am given the following $V = \mathbb R^4$ $W = \{(w,x,y,z)\in \mathbb R^4|w+2x-4y+2 = 0\}$ I have to prove or disprove that $W$ is a subspace of $V$. Now, my linear algebra is fairly weak as I haven't taken it in almost 4 years but for a subspace…
Future Math person
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Determine which of the following mappings F are linear

I'm having a really hard time understanding how to figure out if a mapping is linear or not. Here is my homework question: Determine which of the following mappings F are linear. (a) $F: \mathbb{R}^3 \to \mathbb{R}^2$ defined by $F(x,y,z) = (x,…
user6886
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