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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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Subspace of $M_2(\mathbb{R})$

Let $V = M_2(\mathbb{R})$ and $W = \lbrace A \in M_2(\mathbb{R}) : A = A^T\rbrace$. Determine if $W$ is a subspace of $V$. I am a bit confused about the vector space of $2\times 2$ matrices and how to start over with the transpose function.
Jennah Lee
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Does exists a matrix $B$ such that $A^TA=A^TB+B^TA$? with $B^TB$ being a diagonal matrix and $A$ an incidence matrix

$A$ is a incidence matrix for some undirected graph. $A^TA$ is a positive definite matrix, so I know that we can factorize it as $A^TA = C + C^T$ There exists always a matrix $C$ such that $C = A^TB$? Satisfying the next requirement, $B^TB = cI$ is…
user51196
4
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Projection of matrix onto subspace

I am confused about why the orthogonal projection of matrices onto subspaces is given by a change-of-basis-like formula. For example, in the below image from these notes, why is the orthogonal projection of matrix A onto the subspace $V_m$ given by…
Physics Enthusiast
  • 524
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Relation between Ax=0 and Ax=b

I'm not sure how to solve this. My teacher told me that if $A\vec{x}=0$ has one unique solution (the trivial) then $A\vec{x}=\vec{b}$ has only one unique solution. But I don't know how to prove this.
Mathomat55
  • 547
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vector space generated by $\{I,A,A^2,\dots,A^{2n}\}$

$A$ be $n\times n$ matrix then the dimension of vector space generated by $\{I,A,A^2,\dots,A^{2n}\}$ is atmost $n$ right? as $c_0I+c_1A+\dots+c_nA^n=0$ with some nonzero co efficient(cayley hamilton). so $\{I,\dots A^n\}$ will be linearly…
Myshkin
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4
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prove - For symetric matrix with zero diagonal there exists $UU^t$ s.t $A+2I=UU^t$

I have seen this in a book I am reading it seems that this is straight forward, but unfortunately my algebraic skills not good enough. $A$ is symmetric with zeros on its diagonal, we need to prove that there exist matrix $U$ s.t $$ A+2 I=U U^{T} $$
misha312
  • 537
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The relation between rational forms and Jordan forms.

Is there any algorithm / method that allows one to determine the Jordan form of a matrix after determining its rational form?
user10444
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4
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2 answers

Let $V$ be an inner product space and $W$ be a dense subspace of $V$ then $W^{\perp }={0}$.

Let $V$ be an inner product space and $W$ be a dense subspace of $V$ . Claim: $W^{\perp }={0}$. Let non-zero $\ x\in W^{\perp} \implies (\forall w \in W,\ \langle x ,w\rangle=0)\ \implies W \subseteq x^{\perp}$ and $x^{\perp}$ is closed subspace. So…
RUPAM
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Unsure of attempt to determine $\dim(W_1+W_2)$ and $\dim(W_1 \cap W_2)$

Let $V=\mathbb{R}^4$. $W_1$ is a subspace of $V$ spanned by vectors $a_1=(1, 2, 0, 1)$ and $a_2=(1,1,1,0)$. $W_2$ is a subspace of $V$ spanned by vectors $b_1=(1,0,1,0)$ and $b_2=(1,3,0,1)$. Determine $\dim(W_1+W_2)$ and $\dim(W_1 \cap…
user4167
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Prove: $ T ◦ S $ is surjective iff $ $ T is surjective and $ ImS + KerT = V $

Let $ S: U \rightarrow V $ and $ T: V \rightarrow W $ be linear transformations over field $ F $. Prove that $ T ◦ S $ is surjective iff $ $ T is surjective and $ ImS + KerT = V $ My attempt: $ \rightarrow $ : Suppose that $ T ◦ S $ is surjective.…
hazelnut_116
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Coset of W containing $v$ is a subspace of $V$ iff $v \in W$

I would like if someone could look over my proof. It feels odd to me. Let $W$ be a subspace of a vector space $V$ over a field $F$. Prove that $v + W = \{v + w \mid w \in W\}$ is a subspace of $V$ if and only if $v \in W$. Proof: ($\Rightarrow$)…
user70962
4
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Prove that $A$, a matrix of rank $3$, can't have characteristic polynomial of $p(x) = x^7 - x^5 + x^3$

Prove that $A$, a matrix of rank $3$, can't have characteristic polynomial of $p(x) = x^7 - x^5 + x^3$ My attempt to contradict: Because of that characteristic polynomial, the matrix must be a $7 \times 7$ matrix. Also, $-\mathrm{tr}(A) = 0$,…
Din
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Projection and inner product space

Definition: Let $V$ be vector space, and $U$, $W$ be two subspaces such that $V=U\oplus W$. We know that there exists for each $v \in V$ only one $u \in U$ and only one $w \in W$ such that $v=u+w$. Using this, we define a projection $P_{U,V}\colon…
wantToLearn
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Solving an equation for an unknown matrix

I have come across the following equation but not sure how to solve it for matrix $D$. Let matrices $E$, $M$ and $D$ are three symmetric, $2\times 2$ and semi p.d. matrices. How to solve the equation $M = ED + DE$ for D, where $M$ and $E$ are…
David
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Finding $x$ with $\langle\, Ax, x\rangle > 0$ when matrix $A$ has at least one eigenvalue with positive real part.

Given a real matrix $A_{n\times n}$ with at least one eigenvalue having positive real part, how can I find $x\in\mathbb{R}^n$ such that $\langle\, Ax, x\rangle > 0$? I tried to use the real part of the eigenvector, but it didn't quite work. With…
JPMarciano
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