# 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. In other words, these are the spaces where we can make sense of linear combinations.

12,105 questions
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### Find basis to sub vector spaces

V = $\Bbb R_3 [x]$ and W,U $\subseteq$ V are sub vector spaces. U=$span${$1 - x, x^2, x^2-x^3, -1+x-x^2+2x^3$} W={p(x)$\in\Bbb R_3 [x]$ | p(1)=0 ^ p(2)+p(0)=0} Find basis to W, U+W, U$\cap$W
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### Self-adjoint operator is diagonalisable

I am revising adjoints for a linear algebra exam and am confused as to how to prove this. Suppose that $T: V \rightarrow V$ has the property that $T^*=aT$ for some complex a. How then do you prove ...
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### Basis to vector space with all inverse matrices

$V$ is sub vector space of $M$(2x2) $(\Bbb R)$, V= { $\begin{matrix}a+b & a+e & \\ c-2d&4c-8d\\\end{matrix}$} when $a,b,c,d,e \in \Bbb R$. Give an example of basis to V, that ...
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### What makes a vector superfluous? [on hold]

In Howard Anton's Elementary Linear Algebra -> section 4.3 Linear Independence -> it is said that if we introduce a third coordinate axis in rectangular coordinate system such that the third axis ...
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### Do the following sets span $P_{3}(\mathbb{R})$?

Could somebody please confirm, if I have done this right or whether my approach is right? Consider a $\mathbb{R}$ -Vectorspace $P_{3}(\mathbb{R})$ of real polynomials: $a_{0}+a_{1} X+a_{2} X^{2}$ ...
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### Confusion on the Gram-Schmidt process for complex vectors

I am having some trouble with the inner product and the Gram-Schmidt process for complex vectors as I am trying to learn it on my own. This is mainly due to the discrepancy with my text book and what ...
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### How can this set be a vector space? [on hold]

How can the set {x, y, 0} be a vector space if x+y is not an element of the set? Edit: I mean where x and y are both elements of F.
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### Proof verification for linear subspaces from Michael Taylor's linear algebra notes

Let $V$ be a vector space over a field $\mathbb{F}$ and $W,X\subset V$ linear subspaces. We say $$V=W+X$$ provided each $v\in V$ can be written $$v=w+x,w\in W,x\in X.$$ We say $$V=W\oplus X$$ ...
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### Hamel dimension of a vector space, and dimension of the dual

I have the following (possibly trivial) observation: Let $K$ be an $\mathbb{F}$-vector space (I believe the argument also works for free modules), and let $X\subseteq K$ be it's basis with ...
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### Vector spaces are not rigid

Im following Etingofs Tensor categories and have read about rigid categories now. There it says The category of all vector spaces (including infinite dimensional) is not rigid. Take $V$ to be ...
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### Let $A, B, C$ be finite-dimensional K vector spaces and $f: A → B$ and $g: B → C$ linear mappings. Show the following:

1) $\mathrm{Im}(g ◦ f) ⊆ \mathrm{Im}(g) ∧ \mathrm{Ker}(f) ⊆ \mathrm{Ker}(g ◦ f)$ 2) $rg(g ◦ f) ≤ \min(rg(g), rg(f))$ 3) $\dim(\mathrm{Ker}(g ◦ f)) ≤ \dim(\mathrm{Ker}(f)) + \dim(\mathrm{Ker}(g))$ I'...
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### How are Fn and Finfinity special cases of Fs? [on hold]

If Fn represents the set of all lists of length "n" and Finfinity represents the set of all infinitely long lists, how are Fn and Finfinity special cases of Fs? Here if S is a set, Fs represents the ...
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### Why does the union of $k$-dimensional subspaces contain no „new“ $k$-dimensional subspaces?

For simplification, let $V$ be a finite-dimensional, real vector space. I know that I cannot represent $V$ as the union of finitely many proper subspaces (or even just $k-1$-dimensional subspaces). ...
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### Are all isometries of subsets affine?

I've found at least two questions that deal with whether isometries are affine, Are isometries always linear? and Should isometries be linear? However, both of these questions assume we are ...
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### Is My Understanding Correct

I know there's already some similar questions on this site, but I wanted to phrase my question in a slightly different way. My question is What does "$F^n$ is a vector space over $F$" mean? I ...
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### “vector spaces” without additive inverses

Setup: I know the definition of a vector space: a set $V$ over a field $F$ such that is closed under vector addition and scalar multiplication. My question: Is there a name for spaces that are ...
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### How to do the curl of the product of a function and a vector field

I have the following property which says: $∇ × (f G) = ∇f × G + f (∇ × G)$ Where $f$ is a differentiable function in an open set $S$ on $\mathbb {R}^3$ , and $G$ is a $C^1$ class vector field on $S$ ...
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### Total Variation of Subintervals

Studying functions of bounded variation, the following exercise showed up: Let $I = [a,b]$ be an interval, $(E, \| \cdot\|)_E$ a normed vector space and $f \in BV(I, E)$ a function which is ...
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### Linear Algebra Scalar and Vector Projection [on hold]

I have a problem with one part of a question. I just need help with part b. How would I go about applying the equation to find the scalar and vector projection of v2 onto v1? Thanks!
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### radius of the inscribed sphere of the pyramid in $\mathbb{R^5}$

How to find the radius of the inscribed sphere of the pyramid in $\mathbb{R^5}$ with the vertex in $(1,0,0,0,0)$, which base is a regular $4$-dimensional simplex, luying in the hyperplane $x_1=0$ with ...
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### If $a^2 = b^2$ then which values $a$ and $b$ are constrained to be? [closed]

I've the following subset of $\mathbb{R}^3$: $$Y= \{(a, b,c)^T | a^2=b^2\} \subset \mathbb{R}^3$$ How can I embed the condition $a^2=b^2$ into the vector? That is, what can I say about $a$ and $b$?...
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### Subspace generated [closed]

Let V be the space of the matrices 2 x 2 on R, and let W be the subspace generated by: Find a base, and the dimension of W. Anyone can help me with this problem?
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### Does the convex cone of monotonic functions on a compact set admit a countable conic generating set?

For any function $f:[0,1]^d\rightarrow\mathbb{R}$ and any $i\in\{1,\dots,d\}$ and $x\in[0,1]^d$, let the functions $f_{i,x}:[0,1]\rightarrow\mathbb{R}$ and $\partial_i f:[0,1]^d\rightarrow\mathbb{R}$ ...
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### Light Reflection in a 3D Plane [Raytracing]

This is a problem that I need to figure out to complete a Java Raytracer Render Engine. Let's say a Light is positioned at (x, y, z) position in a 3D Environment. There's also a metallic object, let'...
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### Choosing vectors for projection

I have a vector $v = a_1\mu_1 + a_2\mu_2 + ... + a_n\mu_n$ where $\mu_i$ are given linearly independent but not orthogonal vectors. I need to choose $k$ vectors from the original set such that when ...
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### Orthogonal projection of the vector $p=(1,0,0,0)$ onto the subspace $W=[(1,-3,0,1),(1,5,2,3),(0,4,1,1),(1,-2,0,4)]$

I got a subspace $W=[(1,-3,0,1),(1,5,2,3),(0,4,1,1),(1,-2,0,4)]$ and I want to make an orthogonal projection of a vector $p=(1,0,0,0)$ onto $W$ and onto the orhhogonal complement of $W$. ...
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### Proof that the vector space of continuous functions from $[0,1]$ to $\Bbb R$ is infinite-dimensional

I want to use the following lemma: A vector space V is infinite-dimensional if and only if there exists a sequence of vectors $v_1,v_2,...$ such that $v_1,v_2,...,v_n$ is linearly independent for ...
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### Explicit description of a column space

In a task I should obtain the explicit description for ColA of a given matrix A. The problem that I have is, that I don't know what explicit means in this context. Do I have to write down the columns ...
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### plane edge intersections embedded in higher dimensional space

Let's say we have some D-dimensional euclidean space, and we have some circles of dimension 0 to D-1 (circle dimensionality meaning the minimum number of vectors needed to fully define it, so a ...
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