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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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Infinity matrix norm is maximum row sum norm

I want to prove that the infinity matrix norm is maximum row sum norm. I've shown that for $\|x\|_{\infty}=1$ $$||Ax||_{\infty} = \max_{i}\left|\sum^n_{j=1}a_{ij}x_j \right| \leq \max_{i}\sum^{n}_{j=1} |a_{ij}|\|x\|_{\infty}= \max_{i}\sum^{n}_{j=1}…
dxdydz
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Exponential function and matrices

How do we prove that the exponential function is a bijection between the set $S_n$ of real symmetric matrices with size $n$ and the set $\Sigma_n$ of real symmetric positive definite matrices with size $n$? Thanks.
user63181
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Dual of $\mathbb{Q}$[x] is not isomorphic to $\mathbb{Q}$[x]

Denote by $\mathbb{Q}$ the set of the rational numbers. Denote by $\mathbb{Q}[x]$ the vector space over $\mathbb{Q}$ of the polynomials with rational coefficients. Denote by $(\mathbb{Q}[x] )^{\star}$ the dual of $\mathbb{Q}$[x] . I am trying to…
math student
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Do these vectors form a basis?

I'm researching a potential algorithm, and I'm hoping that someone can verify my calculations. I have sets of vectors in $\mathbb{R}^6$ that I can use. They have a corresponding value associated with them, but this relation is not necessarily a…
Matt Groff
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Why we only need to verify additive identity, and closed under addition and scalar multiplication for subspace?

In the book Linear Algebra Done Right, it is said that to determine quickly whether a given subset of $V$ is a subspace of $V$, the three conditions, namely additive identity, closed under addition, and closed under scalar multiplication, should be…
JOHN
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For any $2$ x $2$ matrix $A$, does there always exist a $2$ x $2$ matrix $B$ such that det($A+B$) = det($A$) + det($B$)?

For each invertible $2$ x $2$ matrix $A$, does there exist an invertible $2$ x $2$ matrix $B$ such that the following conditions hold? (1) $A + B$ is invertible (2) det($A+B$) = det($A$) + det($B$) I know that for $2$ x $2$ matrices det($A+B$) =…
Ryan
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Why two formulas for a in a linear function are equal

Sorry if this is an easy question for some, but I have tried and failed for a while now, so I need someone else to help me. I want to figure out why $$a=\frac{f(x)-b}{x_1}$$ is equal to $$a=\frac{y_2-y_1}{x_2-x_1}$$ For example in a task where you…
Adrian Fagerland
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Showing that given matrix does not have negative eigenvalues without using the knowledge that it is positive definite.

Let $a,b,c$ be a positive real number such that $b^2+c^2
Shweta Aggrawal
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Deciding which statements about matrix A is true where $A^3+A^2-3A+I=0$

A is $2 \times 2$ matrix. $$A^3+A^2-3A+I=0$$ Decide which statements is true. $\quad$ A) 1 is eigenvalue of A. $\quad$ B) Det(A) is 1. $\quad$ C) $A^{-1}$ exists. $\quad$ D) If B is inverse of A, $B^3-3B^2+B+I=0$. Choices are {A,B}, {A,C}, {C,D},…
nik
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How to show that linear map is invertible?

a) Let $L:V \to V$ be a linear map such that $L^2 + 2L + I = 0$, show that $L$ is invertible. b) Let $L:V \to V$ be a linear map such that $L^3 = 0$, show that $I - L$ is invertible. Here, $I$ is identity mapping. For first part, I know that…
hasExams
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How to get the equation where a circle goes through three points

If I have the equation $ax^2+ay^2+bx+cy+d=0$ how do I get the equation where the circumference goes through the points P = (1,1), Q = (−1,−1) and R = (−1,1) I have it in mind to solve it with a matrix, but the instructions seem confusing to me,…
Juju9708
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If $A+B = AB$ then proving that $AB = BA$?

Even though this has been asked before in main site If $A+B=AB$ then $AB=BA$, still I had a query? If $A,B$ are both $n \times n$ matrix and the entries are from $\Bbb{R}$. If it satisfies $A+B = AB$, then can we say that $A$ and $B$ commute? that…
BAYMAX
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Question from Artin's algebra book

Let $A$ be an $n\times n$ matrix such that $A^r =I$ and $A$ has exactly one eigenvalue ,then $A= \lambda I$. My answe: As $A$ is a $n\times n$ matrix then characteristic polynomial has degree n and also exactly one root so $p(x) = (x-a)^n$ ($p(x)$…
jim
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Prove: $ \operatorname{Ker}(T)^\perp= \operatorname{Im}(T^*)$

Let $T:V\to V$ Prove: $ \operatorname{Ker}(T)^\perp= \operatorname{Im}(T^*)$ If $v\in \operatorname{Im}(T^*)$ so $\exists w\in V:T^*w=v$ but how can I continue from here? If $v\in \operatorname{Ker}(T)^\perp$ what does it say?
newhere
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Let $a,b$ and $c$ be real numbers.evaluate the following determinant: |$b^2c^2 ,bc, b+c;c^2a^2,ca,c+a;a^2b^2,ab,a+b$|

Let $a,b$ and $c$ be real numbers. Evaluate the following determinant: $$\begin{vmatrix}b^2c^2 &bc& b+c\cr c^2a^2&ca&c+a\cr a^2b^2&ab&a+b\cr\end{vmatrix}$$ after long calculation I get that the answer will be $0$. Is there any short processs? Please…
priti
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