Questions tagged [vector-spaces]

For questions about vector spaces and their properties. More general questions about linear algebra belong under the [linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars

This tag is for questions about vector spaces and their properties, as well mappings between vector spaces. More general questions about linear algebra belong under the tag.

A vector space consists of a set of elements called "vectors" and is associated with a field (a set with well-behaved notions of addition, multiplication, subtraction and division) called the "field of scalars". An individual vectors can be multiplied by elements of the field of scalars to produce a new vector in the vector space, and pairs of vectors can be added or subtracted to produce a new vector as well. A full introduction can be found on Wikipedia.

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On kernels of commuting operators in infinite dimensions

Let $X$ be an infinite dimensional vector space, and let $\operatorname{S},\operatorname{T}\in\mbox{End}(X)$ be two operators such that: $\operatorname{T}\operatorname{S}=\operatorname{S}\operatorname{T}$, i.e., the two operators…
Bedovlat
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Finding a basis for the solution space of a system of Diophantine equations

Let $m$, $n$, and $q$ be positive integers, with $m \ge n$. Let $\mathbf{A} \in \mathbb{Z}^{n \times m}_q$ be a matrix. Consider the following set: $S = \big\{ \mathbf{y} \in \mathbb{Z}^m \mid \mathbf{Ay} \equiv \mathbf{0} \pmod q \big\}$. It can…
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Confusion about chain of subspaces

This is part of Problem 11.1.6 of Dummit and Foote. The problem reads Let $V$ be a vector space of finite dimension. If $\phi$ is any linear transformation from $V$ to $V$ prove that there is an integer $m$ such that the intersection of the image…
mathmath8128
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Prove that there is a base of $\mathbb R^4$ made of eigenvectors of matrix $A$

Matrix of linear operator $\mathcal A$:$\mathbb R^4$ $\rightarrow$ $\mathbb R^4$ is $$A= \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1\\ 1 & -1 & 1 & -1\\ 1 & -1 & -1 & -1\\ \end{bmatrix} $$ Prove…
Asleen
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What does a norm of a polynomial space mean?

When talking about polynomial vector space, the following example was provided. A polynomial of degree $n$ in two variables is $$p(X)=\sum_{0\leq k+j \leq n} a_{j,k}x_1^jx_2^k$$ where $k+j=n$ and $a_{j,k} \neq 0$. An example of a degree-two…
Trogdor
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Doubt with vectorial spaces (Basis and dimension)

Good night, i'm working in a problem, i need an basis and the dimension of the space. $a_{1}=(1,0,0,-1),\:a_{2}=(2,1,1,0),\:a_{3}=(1,1,1,1),\:a_{4}=(1,2,3,4),\:a_{5}=(0,1,2,3)$ I make this: $\left[ \begin {array}{cccc} 1&0&0&-1\\ 2&1&1&0 \\…
rcoder
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If every non-zero vectors be the eigenvector of a real matrix $A$, prove that $A$ is the scalar matrix $\lambda I_n$.

If every non-zero vectors in $\mathbb{R}^n$ be the eigenvector of a real $n \times n$ matrix $A$ corresponding to a real eigenvalue $\lambda$, prove that $A$ is the scalar matrix $\lambda I_n$. I have no idea to start with. Please help me to solve…
user1942348
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Is the converse of the Pythagorean Theorem false for complex inner products?

I was thinking about the converse of the Pythagorean theorem: $\lVert x + y\rVert^2 = \lVert x\rVert^2 + \lVert y\rVert^2 \implies x \perp y$ Does this hold if the inner product $\langle \cdot,\cdot\rangle$ is complex-valued and it induces the norm…
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Matrix equivalent to linear maps - sanity check

I'm reading some Linear algebra notes I found online, and am a bit confused about the following: If $U,V$ are finite dimensional $\mathbb{C}$-spaces with bases $(\mathbf{u}_1,\dots,\mathbf{u}_m)$ and $(\mathbf{v}_1,\dots,\mathbf{v}_n)$ then there is…
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For a Vector Space $V = A \oplus B = A \oplus C \implies dim(B) = dim(C) $?

For a finite dimensional space there is no problem. $dim(V) = dim(A) + dim(B) = dim(A) + dim(C) \implies dim(B) = dim(C)$ For an infinite dimensional space it still holds that $dim(V) = dim(A) + dim(B) = dim(A) + dim(C) $ But this can be…
Tom Collinge
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What do you call this equivalence relation? $A \simeq B$ if $A = P^t BP$ for some invertible matrix $P$

If $A, B$ are square matrices with coefficients in some ring, we say that $A$ is similar to $B$ if $A = PBP^{-1}$ for some invertible matrix $P$. Similar matrices represent the same linear operator with respect to different bases. Now another…
D_S
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Differentiation continuous iff domain is finite dimensional

Let $A\subset C([0,1])$ a closed linear subspace with respect to the usual supremum norm satisfying $A\subset C^1([0,1])$. Is $D\colon A\rightarrow C([0,1]), \ f\rightarrow f'$ continuous iff $A$ is finite dimensional? If $A$ is finite…
Julian
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How do I link dimension of a normed vector space with closedness?

Let $X$ be a Normed Vector Space, for any $x\in X$ and $r>0$. Let $W:=\{y\in X : \|y-x\|\leq r\}$ and $S:=\{y\in X : \|y-x\|
José Osorio
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Show that $\pi(Z)$ acts as a scalar over $\mathbb{g}$

Let $(\pi, V)$ be a finite dimensional irreducible representation of $\mathbb{g}$ $V$ is a vector space of homogeneous polynomials in 3 variables of degree d over $\mathbb{R}$ $\mathbb{g}=\begin{bmatrix} 0 & a & b\\ 0 & 0 & c \\ 0 & 0 &…
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Dimension of the subspace of the polynomial ring over $\mathbb R$

Suppose $P_n =\{ f(x) \in \mathbb R[x] : \deg(f(x)) \leq n\}$ and $W = \{ p(x) \in P_n : p(x) = p(1-x) \}$. Find the dimension of subspace $W$. Firstly I am showing that $W$ is a subspace of $P_n$. Suppose $a,b \in \mathbb R$ and $p, p' \in…
Struggler
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