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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Reflection across a line?

The linear transformation matrix for a reflection across the line $y = mx$ is: $$\frac{1}{1 + m^2}\begin{pmatrix}1-m^2&2m\\2m&m^2-1\end{pmatrix} $$ My professor gave us the formula above with no explanation why it works. I am completely new to…
dsd
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Inner Product Spaces over Finite Fields

Inner product spaces are defined over a field $\mathbb{F}$ which is either $\mathbb{R}$ or $\mathbb{C}$. I want to know what happens if we try to define them over some finite field. Here's an example: Let $\mathbb{F} = \{0,1,a,b\}$ be a finite field…
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Difference between sum and direct sum

What is the difference between sum of two vectors and direct sum of two vector subspaces? My textbook is confusing about it. Any help would be appreciated.
Marion Crane
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Understanding isomorphic equivalences of tensor product

I get some big picture of tensor and tensor product by reading their Wikipedia articles, and several questions and answers posted before by others. But I cannot figure out how to show the following isomorphic equivalences: from Zach Conn: For…
Tim
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How to efficiently use a calculator in a linear algebra exam, if allowed

We are allowed to use a calculator in our linear algebra exam. Luckily, my calculator can also do matrix calculations. Let's say there is a task like this: Calculate the rank of this matrix: $$M =\begin{pmatrix} 5 & 6 & 7\\ 12 &4 &9 \\ 1 & 7 &…
cnmesr
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Is every axiom in the definition of a vector space necessary?

Definition: A vector space over a field $K$ consists of a set $V$ and two binary operations $+: V \times V \to V$ and $\cdot: K \times V \to V$ satisfying the following axioms: Commutativity of $+$. Associativity of $+$. Existence of an identity…
David Zhang
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$\mathbb{R}$ and $\mathbb{R}^2$ isomorphic as groups?

Using the axiom of choice, $\mathbb{R}$ and $\mathbb{R}^2$ are equal-dimensional vector spaces over $\mathbb{Q}$ and so are isomorphic as $\mathbb{Q}$-vector spaces thus as groups. This is obvious, however I recently began reading Godement's…
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Relation between rank and number of distinct eigenvalues of a matrix

Let $T : V\to V$ be a linear transformation such that $\dim\operatorname{Range}(T)=k\leq n$, where $n=\dim V$. Show that $T$ can have at most $k+1$ distinct eigenvalues. I can realize that the rank will correspond to the number of non-zero…
Ester
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Finding a unit vector perpendicular to another vector

For example we have the vector $8i + 4j - 6k$, how can we find a unit vector perpendicular to this vector?
adil
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What is the proof that covariance matrices are always semi-definite?

Suppose that we have two different discreet signal vectors of $N^\text{th}$ dimension, namely $\mathbf{x}[i]$ and $\mathbf{y}[i]$, each one having a total of $M$ set of samples/vectors. $\mathbf{x}[m] = [x_{m,1} \,\,\,\,\, x_{m,2} \,\,\,\,\, x_{m,3}…
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Why it is important for isomorphism between vector space and its double dual space to be natural?

I'm reading the book (by A. Kostrikin) on linear algebra and I feel like I'm really missing something about this idea. I understand the formal proofs of: a) isomorphism between vector space $V$ and its dual space $V^*$ b) natural isomorphism between…
lithuak
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Why are real symmetric matrices diagonalizable?

A matrix is diagonalizable iff it has a basis of eigenvectors. Now, why is this satisfied in case of a real symmetric matrix ?
aaaaaa
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(Dis)Prove $\sum_{i=1}^n\sum_{j=1}^n{(|x_{i}-x_{j}|-|y_{i}-y_{j}|)^2}\geq 4$

Let $n\ge 4$ and two vectors $x$ and $y$ in $\mathbb{R}^n$ that satisfy $\sum_{i=1}^{n}{x_{i}^2}=\sum_{i=1}^{n}{y_i}^2=1$ $\sum_{i=1}^{n}{x_{i} y_i}=0$ $\sum_{i=1}^{n}{x_{i}}=\sum_{i=1}^{n}{y_i}=0$ With these conditions, prove or disprove that…
Andreas
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A rank-one matrix is the product of two vectors

Let $A$ be an $n\times m$ matrix. Prove that $\operatorname{rank} (A) = 1$ if and only if there exist column vectors $v \in \mathbb{R}^n$ and $w \in \mathbb{R}^m$ such that $A=vw^t$. Progress: I'm going back and forth between using the definitions…
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Proof for triangle inequality for vectors

Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below. By considering $u$ and $v$ as two sides of a triangle, we can see that the lengths of the third…
alok
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