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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. In other words, these are the spaces where ...

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Find the minimum value of the quadratic form $x^TMx$ and the corresponding $x$, where $M$ is a symmetric matrix

How do you find the minimum value of the quadratic form $x^TMx$ and the corresponding $x$, where $M$ is a symmetric matrix? I have $x^TMx = x^T(I − 2(vv^T)/\|v\|^2)x$, and I don't know how to continue....
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16 views

Finding Matrix given an Equation

I have the following problem to solve, but can't. $X^T(Xb-y)=0$ where $X$ is an unknown matrix in $\mathbb{R}^{nxp}$, $b$ is a known vector in $\mathbb{R}^{p}$, and $y$ is also a known vector in $\...
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33 views

Number Elements of a Vector Space over a Field of mod p [duplicate]

I saw a question that looked interesting to me, but I have no clue how to go about doing it. It says, "Let $V$ be a vector space over the field $F=\mathbb{Z}_p$, where $p$ is some prime number. ...
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1answer
46 views

Let $\mathbb H$ be algebra of quaternions and let $S$ be the group of unit quaternions. What does it mean for a point to be tangent to $\mathcal S$?

Let $\mathbb H$ be the algebra of quaternions and let $\mathcal{S} \subset \mathbb H$ be the group of unit quaternions. Show that if $p \in \mathbb H$ is imaginary, then $qp$ is tangent to $\...
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2answers
34 views

If v ∈ R^n is a nonzero vector, and I ∈ R^n×n is an identify matrix. Prove that M = I − 2(v(v^T)/||v||^2 is symmetric and satisfies M^−1 = M

I thought about showing M = M^T, so M is symmetric. But I don't know how to compute 2(v(v^T)/||v||^2 to find M and M^T. Any idea?
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1answer
19 views

Vector space on finite commutative set with $q$ elements

Let $K$ be a finite field with $q$ elements and $V$ a $k$-vector space of dimension n. Then: (a) V is a finite set with $q^n$ elements. (b) Number of bases $\{v_1,...,v_n\}$ in $V$ is: $...
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26 views

Track object in space and time

I'm currently struggling to solve the following problem: The goal is determine the movement of an object by tracking an object in 2d-space and time (third dimension). I always know the $x$ and $y$ ...
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24 views

Impossible to conclude about limit even if we have a strong inequality

Let's say I have a function $f:E \to E$ where $E$ is an euclidian space. Moreover there is : $\delta > 0$ such that : $$\forall (x,y) \in E^2, \mid ||f(x)-f(y)||-||x-y|| \mid \leq \delta$$ Then ...
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the set of all solutions of the system $ax+by+cz =0, ex+fy+cz=0$ forms a subspace.

To show that the set of all solutions of the system $ax+by+cz =0, ex+fy+cz=0$ forms a subspace. Subtracting we get $x = \frac{e-b}{a-d}y$ and putting this in any one of the equations we get $z= \...
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1answer
23 views

Are the transformation matrices the relation between basis $\beta$ and $\beta'$?

Are the transformation matrices the relation between basis $\beta$ and $\beta'$? We are told to get the coordinates of vector $v = (1,2,1)$ in both basis $\beta$ and $\beta'$, and find what the ...
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2answers
28 views

Find $U + W$ and $U \cap W$

Find $U + W$ and $U \cap W$ for $U =\{\begin{pmatrix}x_1\\x_2\\x_3\\x_4\\x_5\end{pmatrix} \in \mathbb{R}^5 | \left\{\begin{array}{l}x_2=2x_1-x_3\\x_4=3x_5\end{array}\right. \}$, $W=\{\begin{pmatrix}...
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57 views

Spaces and fields

What is over vector space defined field? I guest this question has no answer. One additionally needs to endow the space ($V$) with either norm (Banach space) or scalar product (Hilbert space) in ...
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1answer
12 views

Nearest vector among vectors with a constant average

I meet the following problem: Given $\bar{x}\in \mathbb{R}^n$, and a set of vectors $x\in \mathbb{R}^n$ with average $c$, i.e, $$\frac{\sum_{j=1}^n x_{j}}{n}=c, \ \ \ \ \forall x$$ I want to find ...
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Scalars on a one dimensional field with inner product and non orthogonal basis.

Let $V$ be a linear vector space with inner product, if $T$ is a operator on $V$, how can i prove that $Dim(Im (T))=1$ iff exists $u, w \neq 0$ in $V$, such that $T(v)=<v,u>w$ for all $v \in V$. ...
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24 views

Proof that vectors span a vector space.

Let $A$ be a matrix of $m \times n$ dimensions and $B$ be a matrix of $n \times m$ dimensions such that $AB=I_{m}$. If $v_{1},...,v_{k}$ are vectors spanning $\mathbb{R^{n}}$, prove that $Av_{1},...,...
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2answers
30 views

A vector in a linearly dependent indexed set is a linear combination of the PRECEDING vectors?

A theorem in my LA book is characterized as follows: An indexed set $S=\{\vec{v_1},...,\vec{v_p}\}$ of two or more vectors is linearly dependent if and only if at least one vector is a linear ...
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2answers
47 views

Show that $P_k(x) = \frac {x(x-1)(x-2)…(x-k+1)}{k!}\forall k = 0,1,…,n$ determine a base in $\mathbb{R}_{\leq n}[X]$

Show that $P_k(x) = \frac {x(x-1)(x-2)...(x-k+1)}{k!}\forall k = 0,1,...,n$ determine a base in $\mathbb{R}_{\leq n}[X]$ I tried induction and tried to find all the coefficient for X but didn't work ...
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28 views

Relation between a vector and its rotated version around specific axis.

Sorry for bothering yee, but I really need your help. I am looking for a mathematical representation/justification of the relationship between a given vector in a 3D space with its rotated version ...
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2answers
32 views

Finding a Basis and Dimension of a Matrix

Given $V = M_{n \times n}(F)$ is a square matrix and $W$ is a subspace of $V$ of symmetric matrices $A$ of trace $0$. How do I find a basis of $W$ and $\dim(W)$?
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3answers
42 views

Symmetric Matrix A of Trace Zero

Just wondering what that term means. I'll provide the following context to the problem: That is, the entries of A satisfy $a_{ij} = a_{ji}$ for all $i, j$ and $a_{11} + ... + a_{nn} = 0$
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1answer
27 views

How does one show a set of sentences are models of infinite vectors spaces over F?

I was going through these notes and had the following: where $F$ is a field and $V$ is a group. Note that $\Sigma_{F}$ is the set of sentences whose models are exactly the vector spaces over $F$ (I ...
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1answer
17 views

How can I prove that a linear map from a 1-dimensional space to itself is really just multiplication by some scalar?

Show that every linear map from a 1-dimensional vector space to itself is multiplication by some scalar. More precisely, prove that if the dimension of $V$ is equal to 1, and $T \in L(V,V)$, then ...
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2answers
16 views

Linear independence of finite dimentional vector space for a linear transform raised to a power

Let $V$ be a finite-dimensional vector space over $F, T∈L(V)$. Suppose 0 $\neq v∈V$ and m is a positive integer for which $T^{m-1} v \neq $ 0 but $T^{m} v= 0$. Show that S={$v, T v, . . . , ...
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How to prove that for every $M \subset X$ subspace exists an $N \subset X$ subspace such that $M\oplus N = X$?

Let $X$ be a vector space. How to prove that for every $M \subset X$ subspace exists an $N \subset X$ subspace such that $M\oplus N = X$?
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1answer
35 views

How did I obtain incorrect results for $P_{B\leftarrow C}$ and $P_{C\leftarrow B}$ geometrically?

I'm given two bases in $\mathbb{R}^2$: $B = \{\begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} -1 \\ 2 \end{pmatrix}\}$ and $C = \{\begin{pmatrix} -4 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 5 \...
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1answer
24 views

How do I prove that S is linearly independent?

Let {$v_1$, $v_2$,..., $v_r$} be basis of V and {$w_1$, $w_2$,..., $w_m$} be basis of W. Denote S = {$v_1$, $v_2$,..., $v_r$, $w_1$, $w_2$,..., $w_m$}. Show that S is linearly independent. Suppose I ...
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2answers
63 views

If $\langle T v,v\rangle=\langle v,v\rangle$ for all $v\in V$ satisfying $\langle v,v\rangle=1$, then $T$ is the identity.

I'm trying to prove the following: Let $V$ be a vector space over $\mathbb{C}$ and let $T$ be a linear operator on $V$. If $\langle T v,v\rangle=\langle v,v\rangle$ for all $v\in V$ satisfying $\...
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1answer
24 views

Definition of Clifford Algebra

Cliffard algebra defined by relation: $x*y+y*x=g(x,y)1$, where g(x,y) is bilinear symmetric form. What does mean $g(x,y)1$, why it's not just $g(x,y)$, without the identity?
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21 views

show that $(x_n)_{n\geq0}$ and $(y_n)_{n\geq0}$ are l.i. in $Fib$ => $(x_0,x_1);(y_0,y_1)$ are l.i. in $\mathbb{R}^2$

Let $Fib=\{(x_n)_{n\geq0}|x_{n+2}=x_{n+1}+x_n, x_n\in\mathbb{R}\}$. Determine a base in Fib. My attempt: It is easily to be seen that $Fib$ is a vectorial space over $(\mathbb{R}, \oplus,\odot)$. ...
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31 views

Find base in a $4\times 4$ matrix.

Determine a base in the matrix $A$ and determine dimension of $\ker(A)$ and dimension of $\operatorname{Im}(A)$. With $$A=\begin{pmatrix}1&-2&4&-1\\3&-1&2&-1\\-1&-3&...
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2answers
36 views

Showing a basis of $V$ over $\mathbb C$

Given that $V$ is over $\mathbb{C}$ and $\{u, v, w\}$ are distinct and a basis of $V$. Show that $$\{u - (1 +i)v, u+v+w, -2iu\}$$ is also a basis. So far, I have just started by stating that they ...
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1answer
22 views

Writing a vector as a linear combination of vectors from another basis

I have the bases $B=\{\begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} -1 \\ 2 \end{pmatrix}\}$ and $C=\{\begin{pmatrix} -4 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 5 \end{pmatrix}\}$. I'm asked ...
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1answer
28 views

Is it true that: $(U + W)\cap S = U \cap S + W\cap S$?, where $U,W,S \subseteq V$ where $V$ is a vectorial space.

Is it true that $(U + W)\cap S = U \cap S + W\cap S$?, where $U,W,S$ $\subseteq V$ where $V$ is a vectorial space. My attempt: $$\begin{array}{l}U=\{\lambda(\begin{array}{c}u_1\\\vdots\\u_n\end{...
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18 views

Finding a basis from given coordinate representation [on hold]

Determine a basis in $\mathbb{R}^3$ for which the vector $x = [1, -1, 2]^T$, has the representation $[1, 1, 1]^T$. Is this basis unique? Explain.
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26 views

Calculating Null T and Range T for the linear transformation $T(x,y,z)=(x+2y-z,y+z,x+y-2z)$?

I have a doubt regarding the calculation of range of a linear transformation. I will explain my doubt with an example. Suppose, $T:R^3 \to R^3 \ni$ $T(x,y,z)=(x+2y-z,y+z,x+y-2z)$ is a Linear ...
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1answer
26 views

orthonormal set depending on variable

Let $A(t) = \{x_1(t), x_2(t),..., x_n(t) \}$ with $0 \leq t \leq 1$ where $x_i(t) \in \mathbb{R}^n, \forall i$. I would like to construct a set $A(t)$ such that $A(t) $ is an orthonormal set, i.e. $...
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1answer
36 views

Purely geometric solution for finding $P_{B\leftarrow C}$ and $P_{C\leftarrow B}$

I'm given the bases in $\mathbb{R}^2$: $B=\{\begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} -1 \\ 2 \end{pmatrix}\}$ and $C=\{\begin{pmatrix} -4 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 5 \end{...
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3answers
25 views

Vector spaces dimensions unclear theorem

While studying vector spaces I came across this theorem (see image) which does not make any sense to me. Is it possible that in the last row they meant $\;\{v_i\}^k_{i=1} \;$ or something similar? ...
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15 views

How to prove the relationship between krasnoselskii's genus and the dimenson of a vector space?

I have to work with this definition for Genus: Let us denote by $U$ the class of all closed subsets $A ⊂ X- \{0\}$ that are symmetric with respect to the origin, that is, $u ∈ A$ implies $−u ∈ A$. ...
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1answer
22 views

Basis for vector space of functions from integers to reals subject to averaging condition.

Consider the space of all functions from the integers to the reals subject to the following condition. The value of a function in this space at $n$ where $n$ is any integer is the average of the ...
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1answer
25 views

What would a rigorous proof that the vector space of functions from $\mathbb{Z}$ to $\mathbb{R}$ is not finite-dimensional look like?

So consider the vector space of functions from $\mathbb{Z}$ to $\mathbb{R}$. Intuitively, this vector space is not finite-dimensional. Indeed, $\mathbb{Z}$ and $\mathbb{N}$ are the same by "zig-...
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35 views

Verify that an eigenvector is orthogonal to 2 other eigenvectors

Verify that the eigenvector v3=(4,-2,1) corresponding to the eigenvalue e2=16 is orthogonal to the eigenvectors v1=(1/2,1,0) and v2=(-1/4,0,1) (both) corresponding to eigenvalue e1=-5 All I can ...
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1answer
20 views

How is The polynomials of the form $a+bx^2$ closed under addition?

I was surprised to see the polynomials of the form $a + bx^2$ is a vector space since I was sure it was not closed under addition. Couldn't a term in the polynomial be negative, and when adding ...
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0answers
34 views

Is Piecewise linear function necessarily convex?

I was hearing the lecture videos of 'Boyd' on 'Convex functions'. It says piecewise linear functions are convex. The reasoning it presents is that a piecewise linear function can be thought of as ...
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1answer
19 views

Are $W$ and $L$ Subspaces of $V$ or not.

I solves my homework, And I need to tell me if it is true or not please. My professor is very carful, so if there exist any simple mistake tell me please. (Sorry, I don’t speak English well) Let $V=...
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1answer
14 views

Clarification of summation term in a linear combination

Could someone clarify for me with the summation symbols are needed here? Aren't $\alpha_h$ and $\beta_k$ enough to construct every element of W?
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1answer
26 views

How to find the intersection of $W$ and $Z$? [duplicate]

Subspaces$W$ and $Z$ of $\mathbb R^4$ are generated by $\{(1,1,0,-1),(1,2,3,0),(2,3,3,-1)\}$ and $\{(1,1,0,-1),(1,2,3,4),(0,1,3,5)\}$, respesctively. Find a basis for $W$$\cap$$Z$. I already ...
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2answers
19 views

Even out a numerical curve

I have a set of values that increase / decrease exponentially thus: ...
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1answer
22 views

Some notation for vector space $\mathbb{R}^\mathbb{R}$, $\mathbb{R}^{X}$, $C(X)$

I am reading some slides for functional analysis, and it mentioned that $\mathbb{R}^\mathbb{R}$, $\mathbb{R}^{X}$, and $C(X)$ are all vector spaces. Since the slides are so brief and it doesn't ...