# 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

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### Easy way to write standard basis for $F^n$?

Let $F$ be a field. The vector space $F^n=\{(a_1,\ldots,a_n):a_i\in F\}$ has a standard basis, where each of the vectors in the basis are $(1,0,\ldots,0)$, $(0,1,\ldots,0)$ and so on. Right now, I am ...
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### Generate distant n-dim points of numbers

I would like to randomly generate a set of n-dimensional points such that each point is far from all other points by a distance of $d$. In other words, I would like to have a set $S$ of n-dim points ...
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### If $W \cong U \otimes V$, then what is $W$'s bilinear map?

Suppose that $U, V, W$ are vector spaces over a field $F$. Let $B : U \times V \to U \otimes V$ be the universal bilinear map (any bilinear map exiting $U \times V$ factors through $U \otimes V$ as $B$...
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### Constructing a vector space of 8 elements from the field of 2 elements

I would like to use the field of 2 elements ($\mathbb{Z}/2\mathbb{Z}$) to construct a vector space consisting of 8 elements. I have succeeded in constructing vector spaces of 2 elements and 4 elements ...
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### Dimension of $L(V,W)$ when $V, W$ are infinite dimensional?

For finite dimensional $V$ and $W$, we know that \begin{equation*} \dim{L(V,W)} = \dim{V}\cdot\dim{W} \end{equation*} Does this theorem hold for infinite-dimensional vector spaces $V$ and $W$ too?
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### Correct way to say linear dependence/independence

$S=\{v_1,v_2,\ldots , v_n\}$ is a subset of a vector space $V$ over field $\mathbb{F}$. Which is the correct way to describe $S$? $S$ is linearly dependent/independent or vectors of $S$ are linearly ...
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### How do you prove that the determinant of a matrix is the same as the determinant of a transposed matrix? [closed]

Or differently phrased, how would you go about proving det(A) = det(A') This is as clear as it gets, I don't see why the question was closed. I know that they both have the same results, I am just ...
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### Set of all sequences that have infinitely many coinciding elements? [closed]

Set of all sequences that have infinitely many coinciding elements are subspace of all sequences with real numbers? I am checking every axiom and it holds can you say what axiom does not hold here?
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### All vectors that with given not zero $a$ vector do form a vector space? [closed]

For all set of vectors from origin that form with given non-zero $a$ vector $\alpha$ or $\pi-\alpha$ angle show they form subspace or not? ($0\leq \alpha \leq\pi$) I can't imagine how if we ...
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### What are subspaces of $A$?

Can anyone please tell me what are the subspaces of $A$ ? My Thoughts: $\{\alpha I + \beta J : \alpha , \beta \in \mathbb R , \beta > 0 \}$ is not a vector space and it does not contain any non-...
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### What common definition of norm on the space of analytic functions makes the basis $e_n=\frac{x^n}{n!}$ orthonormal?

I mean, this basis, with factorials is very useful as it is the basis of Taylor expansion. But it is not orthonormal under usual definitions of norm ($\int_a^b \sqrt{f(x)^2}dx$). I wonder, how it ...
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### Construct a matrix given the null space of A

Construct a matrix A with the null space spanned by the vectors $(2, -3, 1, 1, -1)^t, (1,0,-2,1,1)^t, (2,-2,1,0,-1)^t$ and $(-8,3,1,1,1)$ in $R^5$. I have come to the conclusion that the four vectors ...
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### Subspace and its orthogonal in NSBF

Let $V$ a vector space of finite dimension on a field $K$, $W \subseteq V$ a subspace of $V$ and $f$ a scalar product on $V \times V$. Then $V=W \oplus W^⊥$. This proposition is true. I want to know ...
### Finding value of $m$ using vector dot product
Given the acute angle between the vectors $\textbf{a}=\binom{m}{0},\:\textbf{b}=\binom{1}{m}$ is thirty degrees, find the possible value of $m$ if $m$ is real. A simple dot product gives \frac{\...