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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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How do you know a solution space contains the origin?

In a linear system $$x - 2y + 3z = 0 \\ -3x + 7y -8z = 0 \\ -2x + 4y -6z = 0 $$ The solution space is $\{(2s -3t,s,t) \mid s,t \in \mathbb{R}\} = \operatorname{span}\{(2,1,0),(-3,0,1)\}$ How do we know that the plane in $\mathbb{R}^3$ contains the…
TsTeaTime
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Exterior power of dual space

Let $V$ be a vector space with basis $e_1, \ldots, e_n$ and $V^*$ be its dual space with dual basis $e_1^*, \ldots, e_n^*$. Let $k$ be an integer between $1$ and $n$. Why $\wedge^{n-k}V=\wedge^{k}V^*$? Thank you very much.
user
7
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If the product of an invertible symmetric matrix and some other matrix is symmetric, is that other matrix also symmetric?

The thought came from the following problem: Let $V$ be a Euclidean space. Let $T$ be an inner product on $V$. Let $f$ be a linear transformation $f:V \to V$ such that $T(x,f(y))=T(f(x),y)$ for $x,y\in V$. Let $v_1,\dots,v_n$ be an orthonormal…
Unkz
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Importance of the homogeneity assumption in definition of linear map

Let $V$ and $W$ be vector spaces over field $F$. A function $f: V \rightarrow W$ is said to be linear if for any two vectors $x$ and $y$ in $V$ and any scalar $\alpha\in F$, the following two conditions are satisfied: $f(x + y) = f(x) +…
Pawel Kowal
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Cost of Solving Linear System

As most of us are aware the cost for solving a linear system ("exactly") with Gauss Elimination and other similar methods with a few right hand side and where the matrix has no structure is $\mathcal{O}(N^3)$ where $N$ is the system size. I am…
user17762
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Let $A$ be an $n \times n$ matrix over $\mathbb{C}$ or $\mathbb{R}$. Does $\det(e^A) = e^{\mathrm{tr}(A)}$ always hold?

Let $A$ be an $n \times n$ matrix over a field $\mathbb{K}$ where $\mathbb{K} = \mathbb{C}$ or $\mathbb{K} = \mathbb{R}$. Does $\det(e^A) = e^{\mathrm{tr}(A)}$ always hold? If the field is $\mathbb{C}$, this can be easily proven using the Jordan…
mathjacks
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how to find the basis of a plane or a line?

Find a basis for the plane $x-2y+3z=0$ in $R^3$. Then find a basis for the intersection of that plane with the $xy$ plane. Is there a proper/algebraic way of finding the basis of a plane? Just by looking at it a basis could be $(2, 1, 0)$ because…
idknuttin
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Why is the trivial vector space the smallest vector space?

My book (Elementary Linear Algebra by Andrilli) says: The set $\mathcal{{V}}$ = {${\mathbb {0}}$} is a vector space AND is the smallest vector space. Then the book asks why $\mathcal{{V}}$ is the smallest vector space. I have no idea where to…
user197950
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example of a nonempty subset is closed under scalar multiplication but not a subspace

Could anyone provide an example of a nonempty subset $U$ of $R^2$ such that $U$ is closed under scalar multiplication, but $U$ is not a subspace of $R^2$
DISPLAY888
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Why is this theorem also a proof that matrix multiplication is associative?

The author remarks that this theorem, which is basically all about what happens if we compose linear transformations, also gives a proof that matrix multiplication is associative: Let $V$, $W$, and $Z$ be finite-dimensional vector spaces over the…
eca2ed291a2f572f66f4a5fcf57511
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How to calculate a linear transformation given its effect on some vectors

Im not sure if my question is worded very well, but I'm having trouble understanding how to tackle this problem. Let $T\colon\mathbb{R}^3\to\mathbb{R}^2$ be the linear transformation such that $T(1,-1,2)=(-3,1)$ and $T(3,-1,1) = (-1,2)$. Find…
user979616
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Matrix inverse property, show that $(I + uv^T)^{-1} = I - \frac{uv^T}{1+v^Tu}$

Let $u, v \in \mathbb{R}^N, u^Tv \neq -1$. Thereby $I +uv^T \in \mathbb{R}^{N \times N}$ is invertible. Show that: $$(I + uv^T)^{-1} = I - \frac{uv^T}{1+v^Tu}$$ I'm lost, why did the denominator get $uv^T$ as $v^T u$? Where did this $1$ come from?…
Clash
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Prove that if $P(P(x)) = Q(Q(x))$, then the polynomials $P$ and $Q$ are equal.

Let $P$ and $Q$ be polynomials with complex coefficients such that $P(P(x)) = Q(Q(x))$. Prove that $P = Q$. It is obvious that degree of both will be equal. But I don't have any idea how to solve this question.
Satvik Mashkaria
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Linear dependence of linear functionals

Problem: Let V be a vector space over a field F and let $\alpha$ and $\beta$ be linear functionals on $V$. If $\ker(\beta)\subset\ker(\alpha)$, show $\alpha = k\beta$, for some $k\in F$. A proposed solution is in the answers below.
Potato
  • 40,171
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why is the following thing a projection operator?

Let $T: E \rightarrow E$ be an endomorphism of a finite-dimensional vector space, and let $S$ be a circle in the complex plane that does not intersect any eigenvalues of $T$. Now let $Q = \frac{1}{2\pi i} \int_S (z-T)^{-1} \, dz$. Why is $Q$ a…
Dylan Wilson
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