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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what does linearly independent in C[0, 1] mean?

This is a question from my textbook I'm not quite sure what C[0, 1] mean, I tried to google the similar question and found that $C[0,1]$ usually denotes the collection of continuous functions $f: [0,1]\to \mathbb{R}$, but I'm still not quite sure…
whoisit
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Check if two 3D vectors are linearly dependent

I would like to determine with code (c++ for example) if two 3D vectors are linearly dependent. I know that if I could determine that the expression $ v_1 = k · v_2 $is true then they are linearly dependent; they are linearly independent…
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How to find $W^{\perp}$ in the following polynomial inner product space?

Consider $P_3(\Bbb{R})$ with inner product $\langle p(x),q(x)\rangle=\int^1_{-1} p(x)q(x)dx$ and let $W=\{ p(x)\in P_3(\Bbb{R})|p(0)=p'(0)=p''(0)=0\}$. How to find $W^{\perp}$? Let's set $p(x)=a+bx+cx^2+dx^3$. Since $p(0)=p'(0)=p''(0)=0$,…
CoolKid
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Why does addition not make sense on infinite vectors?

I was reading http://www.math.lsa.umich.edu/~kesmith/infinite.pdf to learn more about infinite dimensional vector spaces, and the author argues that the standard basis ($e_i$ is the sequence of all zeroes except in the i-th position, where there…
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How to list all possible dimension of $\ker{T},\ker{T^2},...,\ker{T^{k-1}}$ and the corresponding canonical forms?

Let $V$ be $5$-dimension vector space, and $T:\ V\rightarrow V$ a nilpotent linear transformation of order (index) $k$ where $1\le k\le 5$. How to list all possible dimension of $\ker{T},\ker{T^2},...,\ker{T^{k-1}}$ and the corresponding canonical…
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Name for "Almost a vector space, but with $\mathbb{N}_0$ instead of a field"

I have a finite set of vectors $V\subset \mathbb{R}^n$ Let us enumerate $V = \{\tilde{v}_1, \tilde{v}_2,...,\tilde{v}_m\}$ I have some space that I want to talk about (I spend a lot of time talking about and thinking about this space): $L\subset…
Frames Catherine White
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Relationship Between Basis For Vector Space And Basis For Dual Space

There exist the famous theorem about a basis for dual space Let $\mathbb V$ be finite dimensional vector space over $F$ and $\mathcal{B} = \{\alpha_1, \ldots ,\alpha_n\}$ is basis for vector space $\mathbb V$ then $\mathcal{B^*} = \{f_1, \ldots…
Babak Miraftab
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When is $P_\mathcal S ACA'Q_\mathcal S = 0$?

All matrices in this question are real. Let C be a positive definite matrix and $A$ an arbitrary square matrix such that the product $ACA'$ makes sense. Let $\mathcal S$ be a space such that $R(A')\subseteq \mathcal S$, where $R$ here stands for…
KOE
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What does it mean that the standard normal distribution is invariant under orthogonal transformation?

What does it mean that the standard normal distribution is invariant under orthogonal transformation? This is the context where I found that statement: consider $H\subseteq \mathbb{\mathbb{R}^l}$ a $k$-dimensional linear subspace of $\mathbb{R}^l$.…
TEX
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Why is the delta function the continuous generalization of the kronecker delta and not the identity function?

In a discrete $n$ dimensional vector space the Kronecker delta $\delta_{ij}$ is basically the $n \times n$ identity matrix. When generalizing from a discrete $n$ dimensional vector space to an infinite dimensional space of functions $f$ it seems…
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What are some books for infinite dimensional linear algebra?

From the linear algebra books that I've encountered, they either discuss exclusively about finite-dimensional vector spaces, or assume that the reader already knows about infinite-dimensional vector space, Hamel basis, etc. What books explain the…
Henricus V.
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Intersection of the corners of a hypercube and a hyperplane

Consider the corners $c$ of a unit hypercube in $\mathbb{R}^n$ (for example in $\mathbb{R}^2$, $c = \{\{0,0\},\{1,0\},\{0,1\},\{1,1\}\}$) and a hyperplane $p \subseteq \mathbb{R}^{n-1}$ (for example in $\mathbb{R}^{2-1},$ $ p $ is a line). How do I…
Jimmy Xiao
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What is the difference between direct product and direct sum of a finite number of group representations.

I have been reading in Fulton & Harris's book on representation theory and it talks about things like the decomposition of a direct product of representations $ V \otimes V $ into a direct sum of representations. It seems to me there is a real…
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What is the meaning of "unitize a vector"?

The expression "to unitize a vector" is often use in computational geometry. What does it mean?
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Finding ranks and nullities of linear maps

I am confused about ranks, nullities and bases of the kernel. From what I understand the rank is the dimension of a vector space generated by a matrix. How would I do the following examples? Find the ranks and nullities of the following linear maps…
NeoXx
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