For questions about vector spaces and their properties. More general questions about linear algebra belong under the [tag:linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars. In other words, these are the spaces ...

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Few basic things unclear to me about inner product spaces and orthonormal basis

Few things unclear to me about inner product spaces: assume V is an inner product space with B orthonormal basis. Why is it true that: $$\langle x,y\rangle = \langle[x]_{B} , [y]_B \rangle{st}$$ ...
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29 views

Bases of subspaces.

I don't understand how can we prove this. Find a basis of the following subspaces of $\mathbb{R}^4$: a. The vectors $x = (x_1, x_2, x_3, x_4)$ where $x_1=2x_4$ b. The vectors for which $x_1 + x_2 + ...
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2answers
30 views

Going from Linear algebra to Multivariable Calculus [closed]

I just finished a course in Linear algebra, can anyone tell me how Linear and multivariable calc are related?
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25 views

About vector orthogonality with itself and implication in a subspace's complement.

My definition of vector orthogonality is simply that they are if their dot product is $0$. I saw a definition that says The orthogonal complement of a subspace in $\mathbb{R}^n$is the set of ...
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1answer
61 views

Understanding a proof: A set of $m$ orthonormal vectors in $V$, with $m < \operatorname{dim}V$, is not complete.

I read the following proof that in a vector space $V$ of dimension $n$, a set of orthonormal vectors $\{\phi_1, \ldots, \phi_m\}$, with $m<n$, is not complete : Among the linear combinations ...
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2answers
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$T: V \rightarrow V$ a linear transformation such that $T^2 = I$ and $H_1= \{v \in V | T(v) = v\}\ $ and $H_2= \{v \in V|T(v) = -v\}\ $

Let V a vector space and $T: V \rightarrow V$ a linear transformation such that $T^2 = I$ and $H_1= \{v \in V | T(v) = v\}\ $ and $H_2= \{v \in V|T(v) = -v\}\ $ then $V = H_1 \bigoplus H_2$ I stuck ...
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2answers
79 views

For vector x, y, when does |x+y| = |x|+|y|?

In general $|x+y|\le|x|+|y|$. When does equality hold? Spivak "Calculus on Manifolds" says the answer is not "when x and y are linearly dependent." However, that is the answer I get.
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1answer
949 views

How to find rotation angles along X,Y,Z axes with a known vector to bring the axes to correct situation

I am working with 3d point data. When I checked the data I realized that there is some error on my data and need to do some kind of rotational rectification because the points which should be ...
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2answers
32 views

Quotient spaces and quotient groups: equivalence classes and cosets

(Throughout this post, I am talking about vector spaces.) I had the pleasure of doing Abstract Algebra two semesters early, however, I feel like some general context was lost in the process. While I ...
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0answers
35 views

Faster way of finding critical points?

So I am looking at parametric vector function. $$ \begin{vmatrix} \cos (t) & -\sin (t) & 0 \\ \cos f(t) \sin (t) & \cos f(t) \cos (t) & -\sin f(t) \\ ...
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1answer
33 views

Find the signature of $Q(x_1,\ldots,x_n)= \sum_{i,j=1}^{n} a_ia_jx_ix_j$

In $\mathbb{R}^n$ let $Q(x_1,\ldots,x_n)= \sum_{i,j=1}^{n} a_ia_jx_ix_j$ quadratic form. $a:=(a_1,\ldots,a_n)\neq0$ $\in \mathbb{R}^n$ find the signature of $Q$
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1answer
51 views

$E$ an $n$-dimensional vector space. Find all endomorphisms $f$ of $E$ which satisfy $f\circ f = \operatorname{Id}_E$.

Let $E$ be a vector space of dimension $n$. Find all endomorphisms $f$ of $E$ which satisfy $f\circ f = \operatorname{Id}_E$. Is trivial that $f = \operatorname{Id}_E$ is a solution, but I don't ...
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1answer
23 views

If a vector fulfils one condition in this theorem, does it automatically fulfil both?

I have this theorem: If $W$ is a subspace of $\mathbb{R}^n$, for any $x\in \mathbb{R}^n$ there will exist some unique $y\in W$ such that $(x-y)\perp u \ \ : \ \ \forall u \in W$ ...
2
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1answer
26 views

Understanding the orthogonal complement of a subspace.

This is my definition of orthogonal complement: Given a vector subspace if $\mathbb{R}^n$, its orthogonal complement is the set of all vectors in $\mathbb{R}^n$ that are orthogonal to any ...
1
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2answers
88 views

maximum and minimum dimension of the space generated by $\{v_1,v_2,v_3,v_4\}$

I'm confused about this problem. I have four vectors $v_1 = (1,1,1,a), v_2 = (1,2,3,a), v_3= (b,1,0,1), v_4 = (0,b,0,0)$ with $a,b$ real numbers. Determine the maximum and minimum dimension of the ...
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1answer
31 views

Coordinate vector of a subspace of $\mathbb{M}_{2,2}(\mathbb{R})$

Have $$\left\{ \left( \begin{matrix} x & y \\ y & x + y \end{matrix} \right) : x,y,\in \mathbb{R}\right \}$$ Which is a vector subspace of $\mathbb{M}_{2,2}(\mathbb{R})$. I was asked ...
2
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0answers
24 views

About an orthogonal complement theorem

Let $W$ be a subspace of $\mathbb{R}^n$. For any vector $x \in \mathbb{R}^n$, there will one unique vector $y \in W$ that fulfils: $$(x-y) \perp w \ \ : \ \ \forall w \in W$$ I have trouble ...
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2answers
33 views

Determining that this basis is linearly independent with a variable

Have the basis $$B = \{ (1,2,0) , (1,1,1) , (1,a,0) , (0,0,a) \}$$ Explain why doesn't this basis have a dimension of $4$. The only way would be, I guess, that it is linearly dependent, ...
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1answer
35 views

Left shift operator $L: l^2 \rightarrow l^2$ on the sequence space $l^2$

$$L: l^2 \rightarrow l^2$$ is defined by $$b = (b_1,b_2,...) \mapsto Lb = (b_2,b_3,...)$$. $(Lb)_n = b_{n+1}$ respectively. How can I determine the adjoint endomorphism $L^*$? Kind regards George
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2answers
22 views

restricting invertible maps to get new maps

For V and W as vector spaces, let we define V ⊗ W and suppose T be a invertible linear map from V ⊗ W to itself with special condition, I want to know whether there exist something like restricted ...
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1answer
44 views

Finding loci of possible points satisfying vector simultaneous equations

I recently had an exam and a question came up which I was only partially able to answer. The question was the following: Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be constant vectors in ...
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4answers
88 views

Geometrically, what is the span of vectors?

Simple question from a calc 3 beginner. Visually I cannot imagine the span of two vectors, what does this necessarily mean? For example my text mentions if two vectors are parallel their span is a ...
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1answer
26 views

Linear independent vectors of nilpotent transformation

$V$ is a vector space. $N$ is a nilpotent transformation $N:V\rightarrow V$ such that $N^k=0$ ($k$ is the lowest). $v \in V$, $v \notin \text{ker}\ N^{k-1}$ (in other words: $N^{k-1}v \ne 0$). Let ...
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1answer
34 views

Is $\mathcal{L}^p \subset \mathcal{L}^{p-1} $?

A random variable $X$ is called integrable if $E[X] < \infty$. We say that $X \in \mathcal{L}^1$ if $E[X] < \infty$, and in general $X \in \mathcal{L}^p$ if $E[|X|^p] < \infty$. I know that ...
2
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1answer
39 views

2x2 symmetric matrix is a subspace of vector space.

Can you kindly check my proof of the problem and correct if possible. The following $S=\{A\in M_{2,2} | AA^T=A^TA\}$ is a subspace of $V=M_{2,2}$ all real $2\times2$ matrices. My proof: S ...
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1answer
12 views

Given two sets of vectors, is there a relationship that describes whether one of them is “orthogonal” to another?

We saw this theorem regarding orthogonal vector subspaces: Have $$A = \{a_1,a_2,a_3,...,a_k\}\\ B = \{b_1,b_2,b_3,...,b_r\}$$ Bases of vector subspaces $S$ and $T$ respectively. Then: ...
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1answer
22 views

What does the $t$ in $(x,y,z)^t$ mean?

Just a question on notation. I have seen a plane defined this way: $$S = \{(x,y,z)^t \in \mathbb{R}^3 \ / \ 2x-3y+z = 0\}$$ See the $t$ superscript on $(x,y,z)$? Well, I am not quite sure what is ...
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0answers
23 views

q-analog of vector space dimension

I am reading about "quantum dimension" $\dim_q V$ where $V$ is a vector space. In fact, you could write it $[\dim V]_q$ where $\dim V$ is the dimension of the vector space and $[n]_q = ...
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3answers
19 views

How to find the vector that passes through a point and is perpendicular to another vector.

Let $ \mathbb{a} = i+4j-3k$ and $b = 7i+20j-12k$ be vectors and $A(2,5,-3)$ be a point. I want find the line $l_ 3$ passing through point $A$ wich is perpendicular to both veotors. How should I do ...
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2answers
81 views

Way to show curves intersect each other through derivatives and vector question.

!->a!=a,then find the value of-: (all a's are vectors and i,j,k are unit vectors )(! is modulus). !a x i!2+!a x j!2+!a x k!2. Can you also suggest something for-:show that the curves xy=a2 and x2 +y2 ...
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1answer
14 views

Eigenvector shared by two endomorhisms

I am guessing if the following fact is true: Let be $V$ a finite vector space above a field $K$. Let $f, g$ be two endomorphisms of $V$ with $f g = g f$. We assume that both $f$ and $g$ have got at ...
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0answers
19 views

Existence of linear/affine subspace for a number of vectors

Let $V$ be a vector space over a field $K$. Let $k \le \dim V$ be a natural number. I want to show that for each k vectors $v_1, ..., v_k$ there is a linear subspace $U$ of $V$ which has dimension ...
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3answers
38 views

Why does linearly independent spanning set imply minimal spanning set for a vector space?

Suppose β is a linearly independent spanning set of some vector space V. Why must it be the minimal spanning set? In other words, why can there not be two linearly independent spanning sets of a ...
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2answers
133 views

Effect of doubly stochastic matrix on vector norm

Let $D$ be a $N \times N$ doubly stochastic matrix, $x$ be a $N$ dimensional vector. What is the relation between $\Vert Dx \Vert_2$ and $\Vert x \Vert_2$? In addition if $\Vert x \Vert_2=1$, what ...
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2answers
21 views

P3 being subspace of vector space?

V = P3 (all real polynomials of degree at most 3) and $S = \{p(x)\in P_3 | x·p'(x) = p(x),\} $ is it a subspace of vector space $V$? Solution: I don't even know is it possible for the equation ...
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3answers
315 views

Why orthogonal basis?

Lets take the $\mathbb{R}^3$ space as example. Any point in the $\mathbb{R}^3$ space can be represented by 3 linearly independent vectors that need not be orthogonal to each other. What is that ...
21
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4answers
558 views

Proving an integer is non-negative by showing there is a vector space with it as its dimension.

The other day I attended a lecture on methods to show whether or not a number is an integer. We were given examples of showing it is the number of ways to count something, and to show there exist ...
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2answers
50 views

Difference between Euclidean space and vector space?

I often hear them used interchangeably ... they are very complicated to make any use of. Wikipedia words: Euclidean space: One way to think of the Euclidean plane is as a set of points ...
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1answer
59 views

Show that Z cannot be turned into a vector space over any field. [duplicate]

Show that Z cannot be turned into a vector space over any field. So, we have 2 cases here. Case 1:lets suppose the charF=P, n does not equal 0, then (1+1+...+1)n=1n+1n+...+1n=n+n+...+n=pn=wchich ...
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1answer
20 views

fields and subspaces

Let F be a field and let V=F^F, which is a vector space over F. Let w be the set of all functions f element of V satisfying f(1)=f(-1). Is W a subspace of V? a. Has the zero vector b. closed under ...
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2answers
115 views

Relationship between the singular value decomposition (SVD) and the principal component analysis (PCA). A radical result(?)

I was wondering if I could get a mathematical description of the relationship between the singular value decomposition (SVD) and the principal component analysis (PCA). To be more specific I have ...
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2answers
18 views

dimension of the vector space using matrices

Let $C$ be an $n \times n$ real matrix. Let $W$ be the vector space spanned by $\{I, C, C^2, \ldots C^{2n}\}$. The dimension of the vector space $W$ is $ 1.\ 2n \hspace{4cm} 2.\ \text{at ...
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1answer
32 views

problems with finding a basis

Given is : $\mathbb{R^\mathbb{R}_f}:=\{ \alpha:\mathbb{R} \longrightarrow \mathbb{R}| \alpha(x)=0, \}, \alpha(x)\ne0$ only at finitely many points.Show that: $\mathbb{R^\mathbb{R}_f}$ is a subspace ...
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1answer
52 views

difficulties with prooving: K is a vector space over Z/pZ

I am trying to solve the followong exercise: Given is K as a field with finitely many elements. i) show that K is a vector space over $\mathbb{F}_p:=\mathbb{Z/p\mathbb{Z}}$, for some special values ...
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1answer
13 views

V- vector space, show the following equations…

Let V be a K-vector space and S,T $\subseteq$ V be any subset. a. Prove the equation $ <S \cup T>=<S>+<T>$ b. Show based on a counter-example proof that the equation $ <S \cap ...
3
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1answer
23 views

Prove is linearly independent

Prove that that the following subset $S \subseteq V$ in the respectively specified $K$- vector space $V$ is linearly independent a. $K=R$, $ V=R[x] $, $S$= {$x^n-x^m| n,m ∈ R,$ n-even, m-odd}
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1answer
26 views

prove it has basis property

Determine the dimension of the following $K$-vector space $V$, by specifying each having a basis and proving they have Basis property. $K=\mathbb{R}, V= \{ (x_1,x_2,x_3) \in \mathbb{R}^3 \mid ...
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1answer
49 views

Proving a strange vector inequality in the euclidean space

It seems to hold the following inequality in an euclidean reference frame $(x,y,z)$: $$\overrightarrow{U}\cdot\overrightarrow{U}\ge\sqrt{2}\left(\Omega_x+\Omega_y\right)$$ where: ...
0
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1answer
33 views

Find subspaces $W$ and $Y$ of $\mathbb{R}^3$ having the property that $W \cup Y$ is not a subspace of $\mathbb{R}^3$.

I'm prepping myself for graduate linear algebra this fall by attempting self-teach myself some of the "basics" of fields, vectors, etc. found in such linear algebra course. I really don't understand ...
0
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1answer
26 views

Intersection of planes

A line perpendicular to the plane $ 3x-5y+4z-11=0 $ passes through the origin. At what point does this normal intersects the plane?