# 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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### Prove that (MB (φ))^k = MB (φ^k)

Let V be a K vector space with base B: = {b1,…, bn} and φ an endomorphism in V with a representation matrix MB(φ). Prove that (MB(φ))^k = MB(φ^k) for k = 1, ..., n applies.
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### Is there a formula to extract the real part of a complex vector?

For a complex number $z\in \mathbb{C}$, the formulas for the real part and the imaginary part the well-known formula: $$\Re[z]=\frac{z+\overline{z}}{2}\\ \Im[z]=\frac{z-\overline{z}}{2i}$$ What is ...
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### Why is $1,a,a^2,…,a^{n-1}$ linearly independent?

I have a basic question about the proof of "Every finite field extension is algebraic". Given the extension $K\subset L$ with $n:=[L:K]$ and $a \in L$, the proof says, that we have a linearly ...
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### Prove that $||x||_{p} \leq ||x||_{q} \cdot d^{(1/p) - (1/q)}$, if $1 < p < q$ for any $x \in \mathbb R^{d}$

Prove that $||x||_{p} \leq ||x||_{q} \cdot d^{(1/p) - (1/q)}$, if $1 < p < q$ for any $x \in \mathbb R^{d}$ How do you prove this using Holder's inequality?
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### Calculate coordinates of polygon's points after changing lenght of one side

I have a list of points (vectors) on the surface, together they form a polygon. Two sides lead from each point, each side has only two points, these sides form the polygon. And my problem is: How to ...
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### Find points where the plane $6x+y+9z=54$ intersects each coordinate axis.

For this I need to find points $(a,b,c)$ for the $x$-axis, $y$-axis, and $z$-axis. I know how to solve this when given two planes. First I'd set $x=0$ and have $y+9z=54$, but with only one plane I ...
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### Relationship between $A \mathbf{1}$ and$A \mathbf{1}^{\perp}$

I have a square matrix $A \in \mathbb{R}^{n \times n}$ and I know that: $$A \mathbf{1} = e_1$$ where $\mathbf{1}$ is the all ones vector and $e_1$ is the first element of the canonical basis. My ...
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### Symmetric functions for multidimensional variables

I have $N$ variables (let's call them $X$) of dimensionality $D$, that I want to symmetrize. For $D=1$ I know I can use e.g. the elementary symmetric polynomials to accomplish this. What if $D>1$...
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### Dimensions of $Pol(\mathbb{Z}_3)$ (polynomial vector space)

How many dimensions does $Pol (\mathbb{Z}_3)$ have, where $Pol (\mathbb{Z}_3)$ is a vector space of polynomial functions with one variable ($f: x \mapsto \sum^n_{k=0} \lambda_kx^k$). My "guess" would ...
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### Dimension of the vector space defined by matrices product

Let $A$ be an $p×q$ matrix of rank a and $B$ a $r×s$ matrix of rank $b$. Please find the dimension of the vector space $M$. My attempt:I first product some elementary matrices on both sides of $ACB$....
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### The norm of a multilinear transformation between finite-dimensional vector spaces is always finite

I am studying n-linear maps from Zorich, Mathematical analysis II, p. 49-53, where the author writes that "it is not difficult to prove that for mappings of finite-dimensional spaces the norm of a ...
I am given an inclusion of vector spaces $L \subseteq V$ and I know that $V^{\vee \vee} = W^\vee$ for some vector space $W$. There are no assumptions of finite dimensionality. Through the inclusion  ...