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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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Find $x$ in $\mathbb{R}^2$ whose cordinate vector relative to the basis $B$

Consider the basis $B$ of $\mathbb{R}^2$ consisting of vectors $\begin{bmatrix}5\\-1\end{bmatrix}$ and $\begin{bmatrix}-7\\-4\end{bmatrix}$. Find $x$ in $\mathbb{R}^2$ whose coordinate vector relative to the basis $B$ is $[x]_B =…
Shua
  • 235
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How to prove that if $\det(A)=0$ then $\det(\operatorname{adj}(A))=0$?

How to prove that if $\det(A)=0$ then $\det(\operatorname{adj}(A))=0$? I have been trying to solve this but I can't use $$\det(A^{-1})=\det \Big(\frac{1}{\det(A)} \operatorname{adj}(A) \Big)$$ because $\det(A)=0$ and $\frac{1}{0}$ is not allowed.
Miguel Alvarez
  • 105
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proof that $\phi\circ\phi=id$ implies the existence of a diagonal matrix

As exam preparation we were trying to proof the following task: Let $V=\mathbb{R}^2$ and let $\phi$ be an endomorphism of $V$ with $\phi \circ \phi = id$ and $\phi \neq id$ and $\phi \neq -id$. Proof that this implies the existence of a basis…
ftiaronsem
  • 1,089
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Subspaces of $\mathbb{R}^{(0,3)}$

I'm working my way through Axler's "Linear Algebra Done Right", and I've run into a proposition that I don't understand. Namely: The set of differentiable real-valued functions $f$ on the interval (0, 3) such that $f'(2)=b$ is a subspace of…
HandsomeGorilla
  • 555
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What are the linear isometries on $R^n$, equipped with the $l_1$ norm?

Which conditions must the matrix entries satisfy, and what would be an interpretation of the row and column sums of the matrix?
Karsten W.
  • 215
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Why do we need "basis" in linear algebra?

I am very curios about why in linear algebra we need different basis, why can't we just have the standard basis and work with that? And how basis is used in computer graphics? thank you in advance.
LiziPizi
  • 2,855
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Prove or disprove: there exists a basis $p_0, p_1, p_2, p_3 \in P_3(F)$ such that none of the polynomials $p_0, p_1, p_2, p_3$ has degree 2

Prove or disprove: there exists a basis $p_0, p_1, p_2, p_3 \in P_3(F)$ such that none of the polynomials $p_0, p_1, p_2, p_3$ has degree 2 This is a repeat of Does there exist a basis $(p_0,p_1,p_2,p_3)\in P_3(\Bbb F)$ such that none of the…
D.C. the III
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Random Binary matrix

This is a question from Strang's "Linear Algebra and its Applications", right in the first chapter (I'm studying it by myself). I couldn't solve it, it isn't in the Solutions Manual, and my research suggests that there shouldn't be a simple solution…
FrancoVS
  • 211
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Decomposition of vector with respect to direct sum of vector subspaces

$U$ and $W$ are two subspaces of vector space $V$. If $U \oplus W = V$, then $\forall v \in V$, there exist two unique vectors $u \in U$ and $w \in W$ such that $v = u + w$. Is the reverse true? That is, if any vector can has such unique…
Linda
  • 295
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Can matrices with dependent columns being QR factorization?

The problem comes from the $18.06$ Linear Algebra by MIT Open Courseware. The answer: I am very confused. According to the definition, Matrix A -> QR means that A has independent columns. BUT it is obviously that the matrix B is singular in the…
stander Qiu
  • 904
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Eigenvalues, singular values, and the angles between eigenvectors

Suppose the $n \times n$ matrix $A$ has eigenvalues $\lambda_1, \ldots, \lambda_n$ and singular values $\sigma_1, \ldots, \sigma_n$. It seems plausible that by comparing the singular values and eigenvalues we gets some sort of information about…
robinson
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How many ways to choose $l$ vectors in $n$-dimensional space such that every $k$-subset is independent

Working in $F_q^n$. How many different ways do we have to choose $l$ vectors such that every subset of size $k$ of them is linearly independent. (Assume n is large) My Progress: For the first k vectors, just keep out of the subspace spanned so far,…
Martin
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3 answers

Let $A,B,C$ be $n \times n$ square real matrices with $ABC=0$. What is the maximum rank of $CBA$?

Let $A,B,C$ be $n \times n$ square real matrices with $ABC=0$. What is the maximum rank of $CBA$? From an old written examination. I've looked at the kernel and range of each matrix for simple cases (like if $n=2$), but how can we generalize to…
mathjacks
  • 3,624
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Which matricies are a product of projections?

I believe that if $A$ is an $n$ by $n$ matrix over $\mathbb{R}$, then $A$ can be written as a product of projections as long as $A$ is not invertible. However, I don't know how to prove this. By projection I mean a matrix $P$ which satisfies $P^2 =…
jwsiegel
  • 1,628
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Proof of the Addition and Scalar multiplication for linear maps

Let $f, g : U\rightarrow V$ be linear maps and $\lambda\in F$. Then the maps $f + g : U\rightarrow V$ and $\lambda f : U \rightarrow V$ are linear. My attempt at the proof for the first statement is as follows: Let $u,z\in U$ and $a\in F$, using…
UniStuffz
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