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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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Orthogonal set vs. orthogonal basis

For some reason my book distinguishes the two names. If a set is an orthogonal set, doesn't that make it immediately a basis for some subspace $W$ since all the vectors in the orthogonal set are linearly independent anyways? So why do we have two…
mim
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$|\det (A+B)|\ge |\det B|$ for all $B$ such that $AB=BA$ iff $A^2=O$

Let $A\in M_2(\mathbb{C})$. $Z(A)$ is the set of all $B\in M_2(\mathbb{C})$ such that $AB=BA$. Prove that $|\det(A+B)|\ge |\det B|$ for all $B\in Z(A)$ if and only if $A^2=O$. If $A^2=O$ and $A\neq O$, suppose $\lambda$ is an eigenvalue of $A+B$.…
Tien Kha Pham
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Rank of $ T_3$ s.t $ T_3 (T_1)=T_2$

Let $T_1,T_2 : R^5 \to R^3$ be linear transformations s.t rank($T_1$)=3 and nullity ($T_2$)=3 . Let $T_3:R^3 \to R^3 $be linear transformation s.t $ T_3(T_1)=T_2.$ Then find rank of $T_3$
ketan
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Linear algebra homework problem involving basis and dual basis.

Please help me get started on this problem: Let $V = R^3$, and define $f_1, f_2, f_3 ∈ V^*$ as follows: $f_1(x,y,z) = x - 2y$ $f_2(x,y,z) = x + y + z$ $f_3(x,y,z) = y-3z$ Prove that $\{f_1,f_2,f_3\}$ is a basis for $V^*$, and then find a basis for…
Mark
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Given $n$ linear functionals $f_k(x_1,\dotsc,x_n) = \sum_{j=1}^n (k-j)x_j$, what is the dimension of the subspace they annihilate?

Let $F$ be a subfield of the complex numbers. We define $n$ linear functionals on $F^n$ ($n \geq 2$) by $f_k(x_1, \dotsc, x_n) = \sum_{j=1}^n (k-j) x_j$, $1 \leq k \leq n$. What is the dimension of the subspace annihilated by $f_1, \dotsc,…
user169845
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Proving $A^2 = 0$ given $A^5 = 0$

I have a class question where I must prove $A^2 = 0$ given $A^5 = 0$ with A being a 2x2 matrix. I though that I could simply say that as $A^5 = 0$ then $A^2 \cdot A^3 = 0 \implies A^2 = 0$ as $A^2 = A\cdot A$ and $A^3 = A\cdot A\cdot A$ $\implies…
user
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Let $M(n,\mathbb R)$ denote set of all $n\times n$ matrices over $\mathbb R$.Which are true:

Let $M(n,\mathbb R)$ denote set of all $n\times n$ matrices over $\mathbb R$.Which are true: 1.If $A\in M(2,\mathbb R)$ is nilpotent and non-zero ,then there exists a matrix $B\in M(2,\mathbb R)$ such that $B^2=A$ 2.If $A\in M(n,\mathbb R)$ is…
Learnmore
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Prove that $ AA^T=0\implies A = 0$

Let $A$ be an $n \times n$ matrix with real entries, where $n\geq2$. Let $AA^T = [b_{ij}] $, where $A^T $ is the transpose of $A$. If $b_{11} + b_{22 }+\cdots+ b_{nn} = 0$, show that $A = 0$. From what I've gleaned so far, $AA^T$ is a…
Waffleboy
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Complex projections order in inner product

So the complex projection is defined as $$\operatorname{proj}_\vec{u} \vec{v} = \frac{\langle \vec{v},\vec{u}\rangle}{\langle\vec{u},\vec{u}\rangle} \vec{u}$$ with complex inner product. I was wondering if there is a reason why we have to compute…
lllll
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How do I show that the derivative of the path $\det(I + tA)$ at $t = 0$ is the trace of $A$?

Here, I'm taking $A$ to be a linear operator on $\mathbb R^n$ for $n>1$. Can you please tell me how to solve such a problem?
adrija
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Solution of system of three variables

On solving $$ 2x - 4y + z = 0 $$ $$ x + y - 4z = 0 $$ $$ x - y - z = 0 $$ I get $$ y = 0.6 x $$ $$ z = 0.4 x$$ I thought that there was a rule of thumb that you need as many independent equations as the number of unknowns. So, why don't I get…
Leponzo
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Commuting matrices implies upper triangular simultaneously

Let $A_\alpha$ be a family of commuting matrices, that is, $A_\alpha A_\beta=A_\beta A_\alpha$. Show that there exists an unitary matrix $U$ such that $U^*A_\alpha U$ is upper triangular for each $\alpha$. I know that there should be something…
xldd
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How to find a vector space V and decompositions $V=A\oplus B = C\oplus D$ with $A$ isomorphic to $C$ but $B$ is not isomorphic to $D?$

I've tried to solve the following question (Exercise 10, page 107 from Roman's book: Advanced Linear Algebra), but I wasn't able to solve it. Find a vector space V and decompositions $V=A\oplus B = C\oplus D$ with $A$ isomorphic to $C$ but $B$ is…
user23505
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Non trivial solutions for homogeneous equations

Consider the following homogeneous equation where $A$ and $X$ are matrices. $$AX = 0$$ I want to know whether there are non trivial solutions for this equations. Now, if $A^{-1}$ exists, then I can multiply throughout by it and get $X = 0$, so if…
Toiler
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Why is a linear equation with three variables a plane?

In linear algebra, why is the graph of a three variable equation of the form $ax+by+cz+d=0$ a plane? With two variables, it is easy to convince oneself that the graph is a line (using similar triangles, for example). However with three variables,…
russell11
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