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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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vector space axioms imply this?

Consider the vector space axiom $$ r(sv)=(rs)v$$ It is compatibility of scalar multiplication with field multiplication. Here $v$ is a vector in a finite dimensional vector space over a field $F$ and $r,s$ are elements in $F$. Sometimes the…
blue
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For $f : V \to V$ a nilpotent endomorphism, with minimal polynomial $x^m$. Why $f^{m-1}(V) \subset ker(f)$?

I'm trying to get some intuition behind the following theorem: Let V be an F-vector space, dim $V = n$, and let $f : V \to V$ be a nilpotent endomorphism. Then $V$ has a basis $v_1,v_2,\ldots,v_n$ such that $f(v_j)$ is either zero or $v_{j+1}$. My…
Mussé Redi
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Is there a name for the space of vectors orthogonal to a given vector in $\mathbb{R}^n$?

Given a vector $\mathbf{v} \in \mathbb{R}^n$, the set of vectors in $\mathbb{R}^n$ orthogonal to $\mathbf{v}$, namely $$\{\mathbf{u} \in \mathbb{R}^n: \mathbf{u} \cdot \mathbf{v}=0\},$$ forms a subspace. In fact, it is the null space of the $1…
Rebecca J. Stones
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The dimension of centralizer $\gamma=\{B\in M_n(\mathbb{R}):AB=BA\}$

Let $A$ be a $6\times 6$ matrix with charpoly $x(x+1)^2(x-1)^3$. We need to find the dimension of $$\gamma=\{B\in M_n(\mathbb{R}):AB=BA\}.$$ What is the relation of charpoly of $A$ with dimension of the space? Please give me some hints to proceed.
Myshkin
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Linear Algebra - Linear Independence and Span

Suppose $(v_1 ... v_n )$ is linearly independent in a vector space V, and $w \in V$, if $(v_1 + w,.... v_n +w)$is linearly dependent, then $w \in span(v_1 ... v_n)$. I'm still getting the hang of linear algebra proofs, which have a different "feel"…
Astrum
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Intersection of $n-1$ dimensional subspaces...

Let $V$ be an $n$-dimensional vector space over $\Bbb R$. Show that every one dimensional subspace is the intersection of all $n-1$ dimensional subspaces containing it. I honestly have no clue on this one.
Johnny Apple
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I did an Algebra on a vector equation and got what seems to be a contradiction.

This was an edit on a previos question that I don't think anyone saw. In the previous question, I did algebra to an equation with vectors $\vec u, \vec v \in \mathbb R^2$ where I got a function equal to $\frac{\vec u}{\vec v}$ and was trying to…
Adam Gluntz
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Exercise 6.4.24 from Strang's Introduction to Linear Algebra, 5th Edition (2016), page 347

24 (A paradox for instructors) If $A^TA = AA^T$, then $A$ and $A^T$ share the same eigenvectors (true). $A$ and $A^T$ always share the same eigenvalues. Find the flaw in this conclusion: $A$ and $A^T$ must have the same X and same A. Therefore $A$…
Johnny Apple
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rank of a matrix

First of all I am sorry because I have asked similar kind of question a few days ago.But I still have problem with row reductions when there are letters in a matrix.The question is asking the value of 'a' when the rank of matrix is 1 , 2 , 3 and 4.…
Piril
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prove $\text{Hom}(V,V) \to \text{Hom}(W,V/W)$ is surjective

Let $V$ be a finite dimension $k$-vector space, and $W \subset V$ be the subspace, prove the natrual map $$\text{Hom}(V,V) \to \text{Hom}(W,V/W)$$ is surjective. where it maps $\alpha \in \text{Hom}(V,V)$ by first restriting the domain to $W$ then…
yi li
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What about rotation by 180 degrees?

In Sheldon Axler's Linear Algebra on page 76 he writes: The rotation of a nonzero vector in $R^2$ is obviously never equals a scalar multiple of itself. Why is rotation by 180 degrees not the same as multiplication by $-1$?
Frosty
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Proving $(f^*)^*=f$

I solved it like this : $$\langle (f^*)^*(v),w \rangle=\langle v,f^*(w)\rangle=\langle f(v),w\rangle$$ My lecture notes gave a proof with some more steps. Now i'm not sure, maybe i messed something. here $f^*$ denotes the adjoint of the linear map…
sigmatau
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Any proper subspace of $\mathbb{C}^{n}$ has the following property?

Let $V$ be a proper linear subspace of $\mathbb{C}^{n}$ (viewed as a vector space over $\mathbb{C}$). I want to show that, then there exists $\alpha_{1},\dots,\alpha_{n}\in\mathbb{C}$ (not all zero), such that, for all…
rubikman23
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Eigenvector of matrix

Vectors $(1,0,0),(0,1,1),(1,1,1)$ are eigenvectors of matrix $A$. Prove that vector $(1,2,2)$ is eigenvector of matrix $A$. We have: $$A(1,0,0) = \lambda_1 (1,0,0) \\ A(0,1,1) = \lambda_2 (0,1,1) \\ A(1,1,1) = \lambda_3 (1,1,1) $$ Furthermore I…
Thomas
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Can every basis of a vector space be reduced to the standard basis?

Consider the $n$-dimensional vector spaces $V\in\mathbb{R}^n$. Given a non-standard basis $B$ for the vector space $V$, can it be reduced to the standard basis on $\mathbb{R}^n$?
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