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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Polynomials as a Linear Combination
I have read that the set of all Polynomials Pⁿ are also a set of vector spaces.
And the explain I read, said that apart from following all the properties of a vector space (identity, communtativity, ...
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Convergence of vector sequence when inner product converges to $0$
Let $d\in \mathbb N_{\ge 2}$, $(v_i)_{i\in \mathbb N}$ be a sequence of vectors in $\mathbb R^d$. If $\inf\limits_{N\in \mathbb N} \sup\limits_{i>j>N} v_i\cdot v_j =0$, does $v_i$ converge to $0$...
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Confusion over tensor definition of exterior power of a vector space and exterior algebra
I am new to and currently learning about Tensor Algebra and Exterior Algebra. I am confused about the definition of the exterior power of a vector space $V$, $\textstyle \bigwedge^k (V)$, and the ...
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Determine which of these four sets are subspaces of the space $X$ and which of these subspaces are closed.
Let $X = C[-1, 1]$ be equipped with the usual maximum norm. Let $Y_1 = \{ f \in X \mid f(-1) = f(1) \}$, $Y_2 = \{ f \in X \mid f(-1) < f(1) - 1 \}$, $Y_3 = \{ f \in X \mid \int_{-1}^1 f(x) dx \...
user1341210
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Nature of the Euclidean Norm
I've been re-reading my linear algebra book and a definition is given of the norm of a vector in $\mathbb{R}^n$ to be:
||v|| $= \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$.
For $\mathbb{R}^2$ and $\mathbb{...
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Understanding the implication in linear algebra regarding vectors
Let $V$ be a subspace of $\mathbb{R}^n$ with the usual dot product, and let $\mathbf{z}, \mathbf{w} \in V$ be fixed vectors. If for every $\mathbf{v} \in V$ it holds that $\mathbf{z} \cdot \mathbf{v} =...
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Generalized Rotational Matrix for n-dimensional Euclidean Vector Spaces [duplicate]
$R_{ij}(\theta) := \begin{bmatrix}I_{i-1} & 0 & ... & ... & 0 \\
0 & \cos(\theta) & 0 & -\sin(\theta) & 0 \\
0 & 0 & I_{j-i-1} & 0 & 0 \\
0 & \sin(\...
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Affine Map as a Morphism of Affine Vector Spaces
I've recently took interest in morphism and category theory and I'm amazed how it offers a very general notion. However, I'm struggling to apply this for the affine vector spaces.
I've seen that a ...
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direct sum of general linear space
Let $U_1, U_2, U_3$ be subspaces of a linear space $V$ such that $U_2 \cap U_3 = {0}$ and $U_1 \subseteq U_2$, and $V = U_1 \oplus U_3$. We want to prove that $U_1 = U_2$.
I did the following:
Assume, ...
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Why does the dimension of a subspace equal the dimension of the whole space minus the number of conditions on the set?
let's say we define a subspace $$ W = \{ p(x) \in R_3[x]\,\, | \,\, p(170) = 0\} $$
now it is very hard to determine this subspace's dimension via the condition and to solve all polynomials with ...
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Linear space and vector space correlation
I'm confused with definition of vector space and field.
According to wiki vector_space and field.
Vector space over some field is defined as set of element in $V$ and binary operations that satisfies:
...
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How to interpret $L^2$ norm for functions from $[0,T]\to\mathbb{R}^n$?
I have a function $\alpha \in L^2(0,T;A)$ where $A\subseteq \mathbb{R}^n$.
I understand what it means when $A= \mathbb{R}$, i.e.
$$\Bigg(\int_0^T|\alpha(t)|^2dt\Bigg)^{1/2}<\infty$$
If $A\subseteq \...
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For real matrix $M$ and complex vector $v$, is $M(\operatorname{Re}(v))=\operatorname{Re}(M(v))$? [closed]
Let $M \in \mathcal M_n(\mathbb R)$ be real-valued $n\times n$ matrix, and $v =\operatorname{Re}(v)+ i\operatorname{Im}(v) \in \mathbb C^n$ a complex vector. Does it then hold:
\begin{align}
M(\...
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Is there a finite dimensional vector space over a finite field with exactly two bases?
Is there a finite dimensional vector space over a finite field with exactly two bases?
I searched and found that the answer is NO. But I have an example that $\mathbb{Z}_{3}$ is a 1-dimensional vector ...
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Are the requirements for a field higher than for a vector space? [closed]
My background: finished Linear Algebra 1, learning Linear Algebra 2 as of now.
I find myself appreciating the definition of a vector space and the axioms that it has to satisfy much more, as they make ...
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