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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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Defining a linear map via kernel and image.

Are linear maps defined in a 1-1 manner by setting their kernel and image? In other words, If I have a vector space, and I define a set to be the kernel of my would-be linear map, and another set to be it's image. Would I get a well defined, one…
idan
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Complex matrices with null trace

I'm trying to prove the following: Let $A\in \mathbb{C}^{n\times n}$ be a matrix with null trace; then $A$ is similar to a matrix $B$ such that $B_{jj}=0$ (i.e. it has zeroes on its diagonal). Any ideas? Induction on $n$ sounded feasible but I…
Fernando Martin
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If $XYZ=ZXY$ does $e^Xe^Ye^Z=e^Ze^Xe^Y$?

It is well known that if $X,Y$ are commuting matrices, then their exponential commute: $$XY=YX\quad\implies\quad e^Xe^Y=e^Ye^X.$$ Now, I am wondering if the following generalization holds: Question: If $XYZ=ZXY$, does $e^Xe^Ye^Z=e^Ze^Xe^Y$? Note…
user298576
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Is there a non-singular matrix with $k$ ones on each row?

Let $n$ be a fixed positive integer. I would like to know for what values of $k$ there exists an $n$ by $n$ $0/1$ matrix that is non-singular with exactly $k$ ones per row. Clearly if $k=1$ then the identity matrix is non-singular. Also if $k=n$…
user35671
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range of $T$ equals range of $T^2$ if and only if the intersection of the range and kernel of $T$ is trivial

Let $V$ be a finite-dimensional vector space over a field $F$ and let $T$ be an operator on $V$. Prove $\text{range}(T^2) = \text{range}(T)$ if and only if $\text{range}(T) \cap \ker(T) = \{0 \}$. Proof. Assume $v \in \text{range}(T) \cap \ker(T)…
St Vincent
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Degrees of freedom vs. cardinality of tuples

Sometimes it is said that the number of DoF of a system means how many real numbers have to be used at least to describe the system. But we know from set theory that the cardinality of any tuple of reals is the same as the reals, so all the…
kupak
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If $T$ injective or $T$ surjective, what is the composition $T^\ast T$? (where $T^\ast$ denotes adjoint of linear map $T$)

Let $T:V \to W$ be a linear transformation between inner product spaces. Then $T^\ast: W \to V$ denotes the linear transformation with the property that for every $v \in V$ and $w \in W$, $$\langle T(v),w \rangle = \langle v, T^\ast(w) \rangle.$$ We…
user13451345
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2-Norm of a Submatrix is $\leq$ 2-Norm of Original Matrix

Say $A$ is a submatrix of $B$. How do I prove that the $\|A\|_2 \leq \|B\|_2$? I can easily show this for $\|\cdot\|_1, \|\cdot\|_\infty, $ and $\|\cdot\|_F$ and thought maybe the solution lies in relating the inequalities of these other norms to…
sfTurpin1990
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Help with proof about maximum number of eigenvalues

I'm working my way through Linear Algebra Done Right. To help with one proof, I want to prove the following: Given $\mathbf{V}$, a vector space and $T$, a linear operator on it, then: If $\mathbf{W}_1$ and $\mathbf{W}_2$ are subspaces of…
user4832
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How do I figure out the features of a cube in 4 dimensions?

This is a question from Gilbert Strang's Introduction to Linear Algebra and MIT OCW How many corners does a cube have in $4$ dimensions? How many $3$D faces? How many edges? A typical corner is $(0,0,1,0)$. A typical edge goes to $(0,1,0,0)$. I know…
Cherry_Developer
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Linear Algebra by Hoffman and Kunze, Section 3.4, Exercise Problem 12

I am trying to work out problems from Linear Algebra, by Hoffman and Kunze and came across this problem in the exercise of Section 3.4, I have a difficulty solving the (c) part of the problem. Problem 12 If $V$ is an $n$ - dimensional vector space…
Chameleon
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If $T\in\mathcal{L}(V)$ is diagonalizable then $V = \mathrm{null}\; T \oplus \mathrm{range}\; T$

I am self-studying Linear Algebra using Axler Linear Algebra Done Right 3rd Edition. Usually I use the problems from the 2nd edition because there is a solution manual available whereas the 3rd edition has no solutions at all. But I made a mistake…
peter2108
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Understanding first part of dual basis proof

The textbook I'm reading attempts to proof the following: given $\left\{v_1, \ldots, v_n \right\}$ a basis for a vectorspace $V$ over $K$, there exists a basis $\left\{ \phi_1, \ldots, \phi_n \right\}$ for $V^{*}$ defined by \begin{align*} \phi_i…
Kamil
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Trouble with understanding dual space $V^{*}$.

I've just began the study of linear functionals and the dual base. And this book I'm reading says the dual space $V^{*}$ may be identified with the space of row vectors. This notion seems very important, but I'm having trouble understanding it. Here…
Kamil
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Measure change/similarity between two affine transformations

I have two affine transformations $A_1$ and $A_2$ consisting only of a rotation matrix $R_i$ and a translation vector $\overrightarrow{t_i}$ (all in 3D space): $$A_i = \left[ \begin{array}{ccc|c} \, & R_i & & \vec{t_i} \ \\ 0 & \ldots & 0 & 1…
m.s.
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