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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 to find the left null space from rref(A)

I was working through a problem and was wondering if there was an easier way of finding the basis of the left null space of a given matrix. (For a simple example) Suppose we have a matrix $A = \begin{bmatrix} 1 & 2 & 4 \\ 2 & 4 & 8 \end{bmatrix}$…
bigfocalchord
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Properties of $\det$ and $\operatorname{trace}$ given a $4\times 4$ real valued matrix

Let $A$, be a real $4 \times 4$ matrix such that $-1,1,2,-2$ are its eigenvalues. If $B=A^4-5A^2+5I$, then which of the following are true? $\det(A+B)=0$ $\det (B)=1$ $\operatorname{trace}(A-B)=0 $ $\operatorname{trace}(A+B)=4$ Using…
dekchi
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Simple formulae for matrices

Can someone please explain me how to prove the following formula? $det(I +M) = \exp\;tr\;\ln(I+M)\;\,.$ Here $I$ and $M$ are a $2 \times 2$ identity matrix and an arbitrary $2 \times 2$ matrix, correspondingly. Also, how to derive from the above…
Michael_1812
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Conditions for matrix similarity

Two things that are not clear to me from the Wikipedia page on "Matrix similarity": If the geometric multiplicity of an eigenvalue is different in two matrices $A$ and $B$ then $A$ and $B$ are not similar? If all eigenvalues of $A$ and $B$…
Gabriel Nivasch
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Proof that a given matrix has rank $n-1$

Let $x_1 \cdots x_n$ be real numbers $x_i>1$ such that $$\frac{1}{x_1} + \cdots + \frac{1}{x_n} = 1$$ Is it true that the matrix $$ \left[\begin{matrix} x_1-1 & -1 & \cdots & -1 \\ -1 & x_2-1 & \cdots & -1 \\ \vdots & \vdots & \ddots & -1 \\ -1 & -1…
Chris Taylor
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Why are pivot columns the basis of A?

Why is it that when we determine the pivot columns of an m x n matrix $A$, the pivot columns form a basis for the $Range(A)$? I understand that the pivot columns are linearly independent (the reason why we chose them as the pivot columns), but how…
Jonathan Wang
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Prove $\bigcap_{i=1}^k\ker(f_i)\subset \ker(f)\iff f\in {\rm span}(f_1,...,f_k) $

Let $V$ a $\mathbb{K}$-vector space of finite dimension $n$, with $\{f_1,...,f_n\}$ a set linearly independent of $V^*$ and $f\in V^*$. Prove $\bigcap_{i=1}^k\ker(f_i)\subset \ker(f)\iff f\in {\rm span}(f_1,...,f_k).$ ($\Leftarrow$) Let $f\in {\rm…
rcoder
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If $T$ is an upper triangular matrix, and $TT^{H}=T^{H}T$ , show that $T$ has to be a diagonal matrix

Problem If $T \in \mathbb{C}^{n \times n}$ is an upper triangular matrix, and $TT^{H}=T^{H}T$, where $T^H$ means the Hermitian transpose of $T$, show that $T$ has to be a diagonal matrix. What I Have Done This seems to be obvious, but writing $T =…
Mr.Robot
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Show that $\alpha_1u+\alpha_2v+\alpha_3w=0\Rightarrow\alpha_1=\alpha_2=\alpha_3=0$

Let $u, v, w$ be three points in $\mathbb R^3$ not lying in any plane containing the origin. Would you help me to prove or disprove: $\alpha_1u+\alpha_2v+\alpha_3w=0\Rightarrow\alpha_1=\alpha_2=\alpha_3=0.$ I think this is wrong since otherwise Rank…
Sriti Mallick
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Matrices restricted to a subspace

Let $Q$ be an $n\times n$ stochastic matrix. Let $\mathcal S$ be the following subspace of $\mathbb R^n$: $$\mathcal S:=\left\{x\in\mathbb R^n: \sum_{i=1}^nx_i=0 \right\}\, .$$ In a paper that I'm reading, there is a concept that I do not know:…
Mark
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$V$ is a vector space over $\mathbb Q$ of dimension $3$

$V$ is a vector space over $\mathbb Q$ of dimension $3$, and $T: V \to V$ is linear with $Tx = y$, $Ty = z$, $Tz=(x+y)$ where $x$ is non-zero. Show that $x, y, z$ are linearly independent.
Sugata Adhya
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$A$ is consistent iff the augmented matrix has no pivot in last column.

A linear system of equations is inconsistent (does not have a solution) if and only if there is a pivot in the last column of an echelon form of the augmented matrix. I can understand the if part, that is because $0=1$ is impossible. But how…
Silent
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spectral radius and numerical radius of matrix

Let $\mathcal{H}$ be a complex finite dimensional Hilbert space ($\dim\mathcal{H}=d$). Let $A\in M_d(\mathbb{C})$. How can I show without using spectral considerations that …
Student
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prove change of basis matrix is unitary

As the title, let $(V,\langle,\rangle)$ be a complex inner product space and assume $S_1=(u_1,\ldots,u_n)$, $S_2=(v_1,\ldots,v_n)$ are orthonormal bases of $V$. Prove that the change of basis matrix $M_ IV(S_2,S_1)$ is a unitary matrix. (There is a…
i_a_n
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Definition of a Subspace. When to prove that zero vector is in the set?

I was going through this PDF and was reminded of an issue I always run into as outlined below. A subspace for $R^n$ is any collection S of vectors in $R^n$ such that The zero vector 0 is in S. If u and v are in S, then u+v is in S [closed under…
ChocolateAndMath
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