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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Find an orthogonal vector to 2 vector
I have the following problem:
A B C D are the 4 consecutive summit of a parallelogram, and have the following coordinates
A(1,-1,1);B(3,0,2);C(2,3,4);D(0,2,3)
I must find a vector that is ...
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Vector spaces-span
Let $W$ be the set of all $x=(x_1,x_2,x_3,x_4,x_5)$ in $\mathbb{R}^5$ which satisfy
$2x_1-x_2+\frac{4}{3}x_3-x_4=0$
$x_1+\frac{2}{3}x_3-x_5=0$
$9x_1-3x_2+6x_3-3x_4-3x_5=0$.
Find a finite set of ...
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How can one complete a set to a vector basis?
What are the possible ways of solving next trivial task:
$$
\mathbf{u} =
\left( \begin{array}{c}
1 \\
2 \\
0 \\
\end{array} \right)
\mathbf{v} =
\left( \begin{array}{c}
5 \\
5 \\
2 \\
\end{array} \...
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Calculate Dot Product of 2 3D Vectors
During school today (Yr 12 Maths C), we covered finding the Dot Product (Scalar Product) of 2D Vectors of the form (Magnitude, Theta) using the equation:
$$A.B=|A||...
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How can I find the position vector?
There are two planes intersecting at a line.
Plane 1: $x - 2y + z - 9 = 0 $
Plane 2: $x + y - z + 2 = 0$
There is a point $A = (p, q, 1)$ on the line of intersection.
How can I find $p ~\text{and}~...
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Sequences in Banach spaces [duplicate]
I am very bad with proofs that ask you to show that "$\exists$ something..." because in most of them you have to explicitly show the something. The following question is one such. Any help will be ...
user44069
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A simple question on linear transformations between vector spaces
Let $\phi: V \to W$ a linear transformation between vector spaces and $v_1, \cdots, v_k \in V$. Suppose that the following condition is fulfilled:
$$\phi \left (\sum_{n=1}^{k}c_n\mathbf{v_n} \right) =...
user44069
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Let $T,S :\mathcal P \rightarrow \mathcal P$ be such that $T \circ S$ is identity
I came across the above problem and was trying to solve.Could someone point me in the right direction? Thanks everyone in advance for your time.
user52976
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Topology Vector Spaces [closed]
I am new to topology & vector spaces with rudimentary knowledge. Could any one suggest me a comprehensive book on topology with vector spaces ? I find topology & vector spaces - mathematical ...
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Sets forming orthonormal basis
So the question is:
Which of the following sets of vectors form an orthonormal basis for $\mathbb{R}^2$
$(a) \{(1,0)^T, (0,1)^T\}$
$(b) \{(\frac{3}{4},\frac{4}{5})^T,(\frac{5}{13},\frac{12}{13})^T\}$
...
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Replacing one of the conditions of a norm
Consider the definition of a norm on a real vector space X.
I want to show that replacing the condition
$\|x\| = 0 \Leftrightarrow x = 0\quad$
with
$\quad\|x\| = 0 \Rightarrow x = 0$
does not alter ...
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How to get a new point of a vector when rotated.
I want to obtain the new point of a vector that I rotate like this.
When I rotate them, I have the angle of rotation.
I want to know x and y, it rotates taking the reference point of 0,0
Thanks
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Unit vector and directions
I am having problem understanding vectors. If a unit vector points in the direction of $z$ axis, then what coordinates would it have?
The paper I read says $x$ and $y$ but if it is in the direction ...
user2857
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Prove: If $L \leq X$, $L$ has finite dimension, $M\leq X$ Then $L+M$ is closed.
Prove: If $X$ is a locally convex space, $L \leq X$, $L$ has finite dimension, $M\leq X$ Then $L+M$ is closed.
What I know:
If $L$ is a finite dimensional subspace, then $L$ is closed.
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Determine whether the span of one set of vectors contains the span of another set of vectors
How can I determine whether the span of a set of vectors (such as $\mathrm{span}\{(3, 1), (4,1), (0,1)\}$ contains the span of another set of vector?
EDIT: I realize that my original question was too ...
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V is a vector space such that $V = A\oplus A^\perp$ also $V = A \oplus C$ then can we say that $A^\perp = C$?
I have a vector space $V$ such that $V = A\oplus A^\perp$
i.e. $V$ is a direct sum of its subspace $A$ and orthogonal complement of $A$.
Suppose we also have $V = A \oplus C$
Then can we say that $...
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Vector Projection XY plane
How do I find orthogonal projection of a vector $\vec V_1=(2,3,4)^T$ formed with the points $A(0,0,5)$ and $B(2,3,9)$ on $xy$ plane?
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Understanding the structure of a finite dimensional vector space based on the properties of linear maps to itself
Let $V$ be a finite dimensional vector space over $\mathbb{R}$. What can we say about the dimension of $V$ if we know that there exists some linear map $\phi: V\to V$ such that $\phi^n=-I$, where $I$ ...
user21725
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Finite dimensional vector space with subspaces [duplicate]
Possible Duplicate:
Could intersection of a subspace with its complement be non empty.
Is it possible for a finite dimensional vector space to have 2 disjoint subspaces of the same dimension ?
...
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Can you combine axioms for a vector space?
This is what I wrote...initially I wanted to write that it is false because an axiom is a basic property and wouldn't be so basic if you start combining them.
In the axioms for a vector space, can ...
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What is an honest basis?
In a comment to this question, the commentator stated that
"the monomials form an honest basis for your vector space".
To be honest, I never heard of that. Is this something elementary?
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If a vector n is orthogonal to vectors a and b, is it also orthogonal to any linear combination of a and b?
It makes sense conceptually to me I would just like this verified.
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How to prove the linear dependence of these elements?
Let $V$ be a vector space over a field $K$ and let $A = \{c_{1} , c_{2} , ...., c_{n}\}$ be a basis for $V$. If $m > n$ and $a_{1} , a_{2} , a_{3} ...., a_{m}$ are elements of $V$ , how to prove ...
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Finding the dimension of $B(U):=\{T:V \to V: T$ is linear and $T(U)\leq U\}$ where $U\leq V$ [duplicate]
Possible Duplicate:
Set of linear transformations
Let $V$ be an $n$ dimensional vector space over a field $F$ and $U$ be an $m$ dimensional subspace of $V$. Define
$B(U):=\{T:V \to V: T$ is ...
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Is there any math operation defined to obtain vector $[4,3,2,1]$ from $[1,2,3,4]$?
I mean have it been studied, does it have a name?
Like Transpose, Inverse, etc.. have names.
I wonder if the "inversion" of the components position have a name so then I could search material on ...
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Are $\operatorname{Ker}(T)$ and $W/\operatorname{Im}(T)$ isomorphic?
Given a linear mapping $T:V\rightarrow W$ between the two vector spaces, $\operatorname{Im}(T)$ and $V/\operatorname{ker}(T)$ are isomorphic. I wonder if $\operatorname{Ker}(T)$ and $W/\operatorname{...
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how to calculate the angle in the $x-y, y-z, x-z$ plane given only $3D$ vector direction and magnitude?
Please help me solve this. I have been thinking of all sorts of ways to solve this but can't figure out how :(. OK here's the problem: I am given a three dimensional velocity vector (I know the ...
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Formal power series and vector (sub-)spaces
Let $$\mathbb{C}[[x]] := \{\sum_{n\geq 0} a_n x^n | a_n \in \mathbb{C}\}$$ be the set of formal power series of x and $V$ be the vector space of all series over $\mathbb{C}$.
Let in addition $V_1$ be ...
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Dot Product and Orthogonal Complement
Let V be the vector space of all real-valued bounded sequences. Then for $a,b \in V$ $\langle a,b \rangle :=\sum _{n=1}^{\infty } \frac{a(n) b(n)}{n^2}$ defines a dot product. Find a subspace $U \...
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How to calculate the value of $f=\operatorname{dim}(X)-\operatorname{dim}(X \cap W)$?
Suppose that $V$ is a complex vector space with $dim(V)=n$ and $W:dim(W)=n-1$ is the vector subspace of $V$. Suppose also that $X$ is a vector subspace of $V$ but $X\not\subset W$, that is, $X$ is not ...
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Why does the minimization trick still work for infinite dimensional inner product spaces?
I am currently self-studying Linear algebra done right and have reached the part about the minimization problem
$6.58 $Example Find a polynomial u with real coefficients and degree at most $5$ that ...
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Let the Homomorphism $f$ be given by $f: \text{Pol}_3 \mathbb{R} \to \mathbb{R^2}, f(p(x)) = (p'(0), p(1))$ Find a basis for Kernel and Image of $f$.
Let the Homomorphism $f$ be given by
$$f: \text{Pol}_3 \mathbb{R} \to \mathbb{R^2}, f(p(x)) = (p'(0), p(1))$$
Find a basis for Kernel and Image of $f$.
We can write a third degree polynomial as:
$$ax^...
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Let $V$ be vector space over $K$, $v_1,...v_n$ linearly independent and $v_{n+1}\in V$\Span{$v_1,...v_n$} Show $v_1,...v_n,v_{n+1}$ are linearly indep
Let $V$ be a vector space over $K$ and $v_1, ... v_n$ linearly independent, and $v_{n+1} \in V$ \ Span{$v_1, ... v_n$}. Show that $v_1, ... v_n, v_{n+1}$ are linearly independent.
Any idea how to ...
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Properties of Norm in R^2
I have a question regarding properties of norm in $R^2$:
Does $Norm(x,y) = Norm(y,x)$ in general?
If $a, b, c ≥ 0$ and we have $a≥b$, how to prove $Norm(a,c) ≥ Norm(b,c)$?
These seems to be very ...
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Let $S =\{u_1, u_2,...,u_n\}$ be a finite set of vectors. Prove that $S$ is linearly dependent iff $u_1 = 0$ or $u_{k+1} ∈ span(\{u_1, u_2,...,u_k\})$
Let $S =\{u_1, u_2,...,u_n\}$ be a finite set of vectors. Prove that $S$ is linearly dependent if and only if $u_1 = 0$ or $u_{k+1} ∈ \text{span} (\{u_1, u_2,...,u_k\})$ for some $k\space (1 ≤ k<n)....
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Using tangents to avoid a sphere
I have an eagle travelling with vector equation $\vec E_a = [4,5,7] + t[-0.5,-0.8,1]$ towards a spherical guard radius around another eagle's nest with equation $(x+1)^2 + (y+2)^2 + (z-16)^2 = 4$. The ...
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How to formally define points so that points are distinct from vectors? [closed]
First, we define a vector space $(\mathbb R^n,+,\cdot)$ (where $+$ and $\cdot$ satisfy certain axioms).
Next, we define vectors as elements of $\mathbb R^n$.
Now, how do we formally define points so ...
user1210203
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Show $A \subset B \Leftrightarrow B^{\perp} \subset A^{\perp}$ with $A,B$ subsets of a Hilbert space.
Show $A \subset B \Leftrightarrow B^{\perp} \subset A^{\perp}$ with $A,B$ subsets of a Hilbert space.
The forward direction is easy: Assume $A \subset B$. For any $a \in A$ and $x \in B^\perp$ since $...
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Operator norm $4$ different definitions how to prove that $\sup$ is $\max$ and $\inf$ is $\min$ for the last two?
From what I have understood, all the $\sup$'s and $\inf$'s in the $4$ different definitions of the operator norm can be taken as $\max$'s and $\min$'s.
For a linear map $A\in \mathscr L (V,W)$ between ...
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Can a vector with unordered components exist?
It seems like in order for a vector addition to be commutative, it needs to be defined in a "regular" manner, i.e. by adding matching vector components (because then the commutativity of ...
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Extension of an ($\mathbb{R}$, $\boxplus$) $\mathbb{Z}$-module to a vector space over $\mathbb{R}$
I have a $\mathbb{Z}$-module M := ($\mathbb{R}$, $\boxplus$).
Is there an extension theorem I can use to make M a vector space over the reals?
I know I can extend integer scalar multiplication to the ...
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$u_1, u_2$ is linearly independent if and only if the family $u_1 + u_2, u_1 − u_2$ is linearly independent.
I have this exercise where I want to check my solutions. Can someone help me?
Let $u_1$ and $u_2$ be elements of $V$. If $1 + 1 ≠ 0$ in $K$, then the family $u_1, u_2$ is linearly independent if and ...
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Hyperplanes question in Tu's book on Manifolds
I am trying an exercise from Tu's book "An introduction to Manifolds". Specifically, Subchapter 1.3, exercise 2 (b).
I would like, before continuing with my question, to discuss something ...
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Smallest closed subspace which contains $M$
How can one prove following:
If $M\neq \emptyset$ is any subset of Hilbert space $H$, show that $M^ {\perp \perp }$ is the smallest closed subspace of $H$ which contains $M$.
I know that $M \subset M^ ...
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What does it mean exactly that every finite dimensional vector space is isomorphic to its dual space?
For context, I'm trying to really understand bra-ket notation in QM. I tried a few years ago and IIRC, something like $\langle a|b\rangle$ is just the dot product of vectors $a$ and $b$. Technically,...
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dense set in $c_o(\Bbb{Z})$
Let's consider the sequence space $c_o(\Bbb{Z}) = \{(a_n)_{n \in \Bbb{Z}} : lim_{n \rightarrow \infty} (a_n) = 0\}$ equipped with the usual norm $\lVert.\rVert_{l^{\infty}}$.
For any $k \in \Bbb{Z}$, ...
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Exterior power of modules and alternating maps
If $V$ is some finite-dimensional vector space over some field $\mathbb{F}$, then it is well-known that there is an isomorphism
$$(V^{\ast})^{\otimes n}\cong L^{n}_{\mathbb{F}}(V^{n},\mathbb{F})$$
...
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How to explain that $A\vec{x}=0$ has infinite solutions when $A$ has 100 equations in 100 unknowns, and elimination reduces the 100th eq to $0=0$?
There is a problem in Gilbert Strang's Introduction to Linear Algebra that starts like this
32 Start with $100$ equations $A\vec{x}=0$ for $100$ unknowns
$\vec{x}=(x_1,...,x_{100}$. Suppose ...
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Dimension of vector space of matrices
Suppose $A$ is a real $n\times n$ matrix of rank $r$. Let $V$ be the vector space of all real $n\times n$ matrices $X$ such that $AX=O$. What is the dimension of $V$?
My working:
We need
matrices $X$ ...
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Rewriting all unit vectors of a 2-dimensional complex vector space.
The fact below is used in a post regarding the Bloch sphere.
Let $V$ be a two dimensional complex vector space. The collection of all unit vectors of $V$ can be written as
$$\left\{e^{i\gamma}\left(\...
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