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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168 votes
20 answers
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Calculate Rotation Matrix to align Vector A to Vector B in 3d?

I have one triangle in 3d space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current ...
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  • 1,859
164 votes
8 answers
48k views

Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional?

It would seem that one way of proving this would be to show the existence of non-algebraic numbers. Is there a simpler way to show this?
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131 votes
10 answers
153k views

Physical meaning of the null space of a matrix

What is an intuitive meaning of the null space of a matrix? Why is it useful? I'm not looking for textbook definitions. My textbook gives me the definition, but I just don't "get" it. E.g.: I think ...
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105 votes
3 answers
13k views

Why are vector spaces not isomorphic to their duals?

Assuming the axiom of choice, set $\mathbb F$ to be some field (we can assume it has characteristics $0$). I was told, by more than one person, that if $\kappa$ is an infinite cardinal then the ...
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98 votes
7 answers
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Is the vector cross product only defined for 3D?

Wikipedia introduces the vector product for two vectors $\vec a$ and $\vec b$ as $$ \vec a \times\vec b=(\| \vec a\| \|\vec b\|\sin\Theta)\vec n $$ It then mentions that $\vec n$ is the vector normal ...
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  • 1,791
97 votes
7 answers
104k views

Geometric interpretation of $\det(A^T) = \det(A)$

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns, could someone give a geometric interpretation of the property?
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  • 3,501
90 votes
1 answer
4k views

In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual?

I've read in several places that one motivation for category theory was to be able to give precise meaning to statements like, "finite dimensional vector spaces are canonically isomorphic to their ...
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82 votes
9 answers
23k views

In Linear Algebra, what is a vector?

I understand that a vector space is a collection of vectors that can be added and scalar multiplied and satisfies the 8 axioms, however, I do not know what a vector is. I know in physics a vector is ...
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82 votes
2 answers
34k views

Difference between metric and norm made concrete: The case of Euclid

This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me. This time I am making ...
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  • 8,376
76 votes
3 answers
12k views

Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)

I am trying to understand the differences between $$ \begin{array}{|l|l|l|} \textbf{vector space} & \textbf{general} & \textbf{+ completeness}\\\hline \text{metric}& \text{metric space} &...
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  • 8,376
75 votes
18 answers
470k views

How to find perpendicular vector to another vector?

How do I find a vector perpendicular to a vector like this: $$3\mathbf{i}+4\mathbf{j}-2\mathbf{k}?$$ Could anyone explain this to me, please? I have a solution to this when I have $3\mathbf{i}+4\...
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69 votes
1 answer
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What is the logic/rationale behind the vector cross product?

I don't think I ever understood the rationale behind this. I get that the dot product $\mathbf{a} \cdot \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b}\rVert \cos\theta$ is derived from ...
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  • 1,635
69 votes
14 answers
36k views

What isn't a vector space?

I'm really confused about vector spaces. We're learning about them in Linear Algebra, and my book doesn't give good examples of what a vector space is. I understand sets and vectors, but I don't ...
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63 votes
7 answers
52k views

What is the main difference between a vector space and a field?

In my opinion both are almost same. However there should be some differenes like any two elements can be multiplied in a field but it is not allowed in vector space as only scalar multiplication is ...
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63 votes
14 answers
20k views

Dot Product Intuition

I'm searching to develop the intuition (rather than memorization) in relating the two forms of a dot product (by an angle theta between the vectors and by the components of the vector ). For ...
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61 votes
4 answers
490k views

How to tell if a set of vectors spans a space?

I want to know if the set $\{(1, 1, 1), (3, 2, 1), (1, 1, 0), (1, 0, 0)\}$ spans $\mathbb{R}^3$. I know that if it spans $\mathbb{R}^3$, then for any $x, y, z, \in \mathbb{R}$, there exist $c_1, c_2, ...
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57 votes
9 answers
17k views

A linear operator commuting with all such operators is a scalar multiple of the identity.

The question is from Axler's "Linear Algebra Done Right", which I'm using for self-study. We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a ...
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  • 675
57 votes
9 answers
4k views

If the field of a vector space weren't characteristic zero, then what would change in the theory?

In the book of Linear Algebra by Werner Greub, whenever we choose a field for our vector spaces, we always choose an arbitrary field $F$ of characteristic zero, but to understand the importance of the ...
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  • 7,131
56 votes
7 answers
62k views

How to find a basis for the intersection of two vector spaces in $\mathbb{R}^n$?

What is the general way of finding the basis for intersection of two vector spaces in $\mathbb{R}^n$? Suppose I'm given the bases of two vector spaces U and W: $$ \mathrm{Base}(U)= \left\{ \left(1,1,...
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54 votes
1 answer
91k views

Effect of elementary row operations on determinant?

1) Switching two rows or columns causes the determinant to switch sign 2) Adding a multiple of one row to another causes the determinant to remain the same 3) Multiplying a row as a constant results ...
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  • 3,501
53 votes
6 answers
42k views

What are some examples of infinite dimensional vector spaces?

I would like to have some examples of infinite dimensional vector spaces that help me to break my habit of thinking of $\mathbb{R}^n$ when thinking about vector spaces.
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  • 693
51 votes
6 answers
5k views

What do mathematicians mean by "equipped"?

I am a mathematical illiterate, so I do not know what people mean when they say "equipped". For example, I say that a Hilbert space is a vector space equipped with an inner product. What does that ...
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  • 5,176
51 votes
2 answers
28k views

What kind of matrices are non-diagonalizable?

I'm trying to build an intuitive geometric picture about diagonalization. Let me show what I got so far. Eigenvector of some linear operator signifies a direction in which operator just ''works'' ...
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  • 4,911
48 votes
11 answers
4k views

Why study finite-dimensional vector spaces in the abstract if they are all isomorphic to $R^n$?

Timothy Gowers asks Why study finite-dimensional vector spaces in the abstract if they are all isomorphic to $R^n$? and lists some reasons. The most powerful of these is probably There are many ...
48 votes
3 answers
18k views

Every proper subspace of a normed vector space has empty interior

There is a conjecture: "The only subspace of a normed vector space $V$ that has a non-empty interior, is $V$ itself." (here, the topology is the obvious set of all open sets generated by the metric $||...
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48 votes
2 answers
15k views

How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?

How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$? Here's my attempt: Given a Cauchy sequence $\{q_n\}_{n \in \mathbb{N}}$ in $X/Y$, each ...
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47 votes
3 answers
12k views

Is there a vector space that cannot be an inner product space?

Quick question: Can I define some inner product on any arbitrary vector space such that it becomes an inner product space? If yes, how can I prove this? If no, what would be a counter example? Thanks ...
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  • 6,285
47 votes
7 answers
31k views

How to understand dot product is the angle's cosine?

How can one see that a dot product gives the angle's cosine between two vectors. (assuming they are normalized) Thinking about how to prove this in the most intuitive way resulted in proving a ...
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  • 1,509
47 votes
4 answers
2k views

Pathologies in module theory

Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar ...
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  • 933
45 votes
6 answers
22k views

understanding of the "tensor product of vector spaces"

In Gowers's article "How to lose your fear of tensor products", he uses two ways to construct the tensor product of two vector spaces $V$ and $W$. The following are the two ways I understand: $V\...
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44 votes
6 answers
122k views

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 ...
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  • 465
43 votes
3 answers
63k views

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.
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  • 2,267
42 votes
3 answers
9k views

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:...
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  • 3,101
41 votes
2 answers
14k views

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: $...
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  • 4,612
41 votes
6 answers
3k views

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 $+$. ...
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  • 8,332
41 votes
2 answers
14k views

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 ...
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40 votes
1 answer
1k views

$\mathbb{R}$ 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, ...
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39 votes
11 answers
308k views

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?
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  • 437
39 votes
2 answers
77k views

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]...
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  • 4,115
39 votes
2 answers
47k views

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 ...
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  • 2,909
36 votes
3 answers
61k views

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 ...
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36 votes
4 answers
32k views

Why are real symmetric matrices diagonalizable? [closed]

A matrix is diagonalizable iff it has a basis of eigenvectors. Now, why is this satisfied in case of a real symmetric matrix ?
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  • 2,516
36 votes
1 answer
3k views

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 ...
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  • 1,155
35 votes
2 answers
127k views

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 ...
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  • 3,700
35 votes
3 answers
122k views

Change of basis matrix to convert standard basis to another basis

Consider the basis $B=\left\{\begin{pmatrix} -1 \\ 1 \\0 \end{pmatrix}\begin{pmatrix} -1 \\ 0 \\1 \end{pmatrix}\begin{pmatrix} 1 \\ 1 \\1 \end{pmatrix} \right\}$ for $\mathbb{R}^3$. A) Find the ...
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35 votes
2 answers
1k views

(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 $\sum_{i=1}^{n}{x_{i}^2}=\sum_{i=1}^{n}{y_i}^2=1$ $\sum_{i=1}^{n}{x_{i} y_i}=0$ $\sum_{i=1}^{n}{x_{i}}=\sum_{i=1}^{n}{y_i}=0$ ...
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  • 14.2k
33 votes
3 answers
11k views

What Is An Inner Product Space?

As I've understood it, what I've learned is that the dot product is just one of many possible inner product spaces. Can someone explain this concept? When is it useful to define it as something other ...
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  • 582
33 votes
2 answers
12k views

Proof that two bases of a vector space have the same cardinality in the infinite-dimensional case.

I am having a difficulty setting up the proof of the fact that two bases of a vector space have the same cardinality for the infinite-dimensional case. In particular, let $V$ be a vector space over a ...
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  • 24.2k
33 votes
1 answer
4k views

Why is the inclusion of the tensor product of the duals into the dual of the tensor product not an isomorphism?

Let $V$ and $W$ be vector spaces (say over the reals). There is a linear injection $V^* \otimes W^* \to (V \otimes W)^*$ which sends $\sum_i f_i \otimes g_i \in V^* \otimes W^*$ to the unique ...
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  • 20.9k
33 votes
7 answers
869 views

What is the geometric meaning of this vector equality? $\vec{BC}\cdot\vec{AD}+\vec{CA}\cdot\vec{BD}+\vec{AB}\cdot\vec{CD}=0$

I was doing some exercises for linear algebra. One of them was to prove that for any four points $A, B, C, D \in \mathbb{R}^3$ the following equality holds: $$\overrightarrow{BC} \cdot \overrightarrow{...
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