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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Vector Question Help

A plane is determined by $(x,y,z) = (1,-1,0) + t(1,-1,2)$ and point $p(1,2,3)$. find point of intersection of $(1,4,-1)+s(-6,2,-4)$ with this plane. I tried this: given the data plane equation is:$$ ...
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Why we take transpose of Vector (Displacement Vector)?

I'm trying to understand some equations that involves transpose of vectors (displacement vectors to be precise) Two set of vectors F and G (with i,j) that corresponds to X,Y value in plane and ...
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2answers
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Determining line equation

Find the equation of the line going through the point $(2,-3,4)$ ,and which is perpendicular to the plane $ x+2y + 2z = 13$ So I tried this: the normal of the plane is $(1,2,2)$, random point on the ...
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How to find a transition matrix?

let $a= \{a_1, a_2, a_3, a_4\}$ and $b=\{b_1,......,b_4\}$ and $r = \{r_1,...,r_4\}$ Also, $b_1 = 4a_1$ $b_2 = 8a_1 + 7a_2$ $b_3 = 4a_1 + 4a_2 + 4a_3$ $b_4 = 9a_1 + 5a_2 + 8a_3 + 5a_4$ and ...
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Finding the dimension of subspace span(S)

Problem: Consider the set of vectors $S= \{a_1,a_2,a_3,a_4\}$ where $a_1= (6,4,1,-1,2)$ $a_2 = (1,0,2,3,-4)$ $a_3= (1,4,-9,-16,22)$ $a_4= (7,1,0,-1,3)$ Find the dimension of the subspace $span(S)$? ...
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Question about vector equations of lines and planes

Find the equation of the line going through the point $(2,-3,4)$ ,and which is perpendicular to the plane $ x+2y + 2z = 13$ So I tried this: the normal of the plane is $(1,2,2)$, random point on the ...
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2answers
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Vectors Geometry question

Find the equation of the line going through the point $(2,-3,4)$ ,and which is parralel to the plane $ x+2y + 2z = 13$ So I tried this: the normal of the plane is $(1,2,2)$, random point on the line ...
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1answer
34 views

Uinviersal property of basis of a vector space

Let V be a vector space over a field k. Let B be a subset of V. If any set map from B to any vector space W can be extended uniquely to a k-linear map from V to W. Then B is a basis of V. Can ...
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Vectors Question

I have a question regarding Vectors; Find the equation of the plane perpendicular to the vector $\vec{n}\space=(2,3,6)$ and which goes through the point $ A(1,5,3)$. (A cartesian and parametric ...
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30 views

Question about Hahn-Banach separation Theorem

So here is my question, I am just reading about the Hahn-Banach separtion Theorem and there is one case where a question appeared, namely, Let $X$ be a normed $\mathbb R$ vectorspace and let $A,B$ ...
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12 views

what is the dimension of this subspace for given problem

In a subspace $W=\{[a_{ij}]:a_{ij}=0$ if $i$ is even$\}$ of all $10\times 10$ real matrix, what is the dimension of W?
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20 views

Show the Hermitian matrices, with trace(g*g1,1)=0 form a vector space.

This is a question from an example sheet that I think may have a mistake in it. Show that the set of Hermitian matrices $A \in H_2 (\mathbb{C})$ with Trace$(A\cdot A_{(1,1)})=0$ is a real three ...
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1answer
38 views

If $V=V_1\oplus V_2\oplus\dots \oplus V_n$, then is it necessary that $V_i\cap V_j=\{0\}$?

Say $$V=V_1\oplus V_2\oplus\dots \oplus V_n$$ where $V$ is a vector space and $V_1,V_2,\dots, V_n$ its subspaces. Is it necessary that $$V_i\cap V_j=\{0\}$$ I think it is, for unique representation ...
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1answer
18 views

Expected length of projection of vector

How can be the expected length of projection of vector of length $l$ is $l/\sqrt d$ where d is dimension of underlying vector space. I am using $l_2^2$ norm
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1answer
25 views

Unit ball in space of d dimension

If I have a unit ball in space $R^d$ then in how many dimension space its surface will be represented. I know the answer is d-1 but i am unable to convince myself. can anybody give me some intuition. ...
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1answer
26 views

are there any unordered basis,what is the most basic example???

I have been doing linear algebra and I can't really understand the existence of basis other than ordered basis ,but since ordered basis are taught as special arrangement basis then what are other ...
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1answer
29 views

Showing that some differential equation has an infinite dimensional solution space?

I don't see how to proceed or even where to start to show this thing that I have found: The differential equation $$(\sin x)\frac{dy}{dx} - 2(\cos x)y = 0$$ has an infinite solution space of ...
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1answer
33 views

Uniqueness of the solution to some differential equation.

I'm currently working on the subject mentioned in the title in a very general way. I think I get stuck for a stupid reason but here is my problem : I'd like to show that any solution to the equation ...
3
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0answers
32 views

Given three points in $\mathbb R^3$ that define a plane. Need to find the normal of the plane.

I came across this question and it has been troubling me for a while... A plane in $\mathbb R^3$ that contains three points is defined as $A=(1, 2, 3)$, $B=(0, 1, 4)$, $C=(2, 1, -7)$... I have to find ...
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1answer
85 views

An Algebraic Version of vector spaces

Consider the following set of real numbers $\mathcal{X}=\{1,2,3,\sqrt{2}+1,\pi+\sqrt{2}\}$. Lets consider the set of all linear combinations with integer coefficients of these numbers which I will ...
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29 views

Finding the distance between two vectors

If given the basis of a row space is $\{(1, 0, 1, 1), (0, 1, -1, 1)\}$, how can I use this information to find the vector in this row space that is closest to the vector $(1, 1, -1, 1)$? Please ...
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1answer
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$\frac{d(X'X)}{dX}=?$

Thanks a lot for reading my thread. I am wondering what is the derivative of $X'X$ with respect to $X$? Here $X$ is a vector/matrix, and $X'$ is the Hermitian matrix of $X$; It would be great if ...
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1answer
26 views

$A \oplus C = B \oplus C$ but $A\neq B$

Let $V$ be a vector space, with subspaces $A, B,C$ such that $A\oplus C = B\oplus C = V$. Prove or give a counterexample disproving that $A=B$. I am trying to find a counterexample.
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1answer
22 views

Find a basis for $B/\mathfrak{B}^e$

In the context of algebraic integers, I would like to solve te following problem. Let $A \subset B$ be two rings, $\mathfrak{p}$ a prime ideal of $A$ and $\mathfrak{B}$ a prime ideal of $B$ lying ...
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17 views

Advice on vector equation geometry

I was wondering what a good approach and your technique for learning geometrically what objects are from their vector equations, often involving dot and cross products, are and 'easily' identifying ...
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1answer
18 views

Trouble understanding finite vector spaces and Gaussian coefficent

I have studied linear algebra for 2 months now and i cannot understand a task that i am currently trying to solve. Basically i am trying to find the amount of bases for n-dimensional vector space over ...
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1answer
43 views

How to find the coordinates of intersection points between a plane and the coordinate axes?

Can you please explain what I am supposed to do and why that is true? The equation of the plane is 4x - 3y = 12. Is the z coordinate always zero in this plane or not? I mean, it is, if its the ...
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1answer
40 views

Finding basis of vector spaces

Without proof find the dimension and a basis of the following vector spaces $V$ over the given field $K$. $V$ is the set of all polynomials over $\mathbb{R}$ of degree at most $n$, in which the sum of ...
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1answer
27 views

Show $\langle u,v\rangle \neq \mathbb{R}^2 \Leftrightarrow \exists \lambda \in \mathbb{R}(u = \lambda v)$

$u,v \in \mathbb{R}^2$ are different from $(0,0)$ I have to show that $\langle u,v\rangle \neq \mathbb{R}^2 \Leftrightarrow \exists \lambda \in \mathbb{R}(u = \lambda v)$ I am not sure how to start ...
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1answer
34 views

What is the difference between a module of finite rank and finitely generated module.

R is an integral domain and every module we talk about is an R-module. If a module is finitely generated then obviously every element of the module can be written as finite R-linear combination of the ...
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3answers
54 views

Regarding a Basis for Infinite Dimensional Vector Spaces

In my linear algebra class, during the discussion of vector spaces, our instructor mentioned infinite dimensional spaces, including the polynomial space over Q and the space of all continuous ...
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1answer
20 views

sum of two matrices question given condition

How can it be proved that two matrices being orthogonally diagonalizable indicates that their sum is also?
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1answer
29 views

Explain why each set is NOT a basis for the given vector space

My biggest problem with linear algebra is trying to get the wording right when I answer questions. I want to communicate my answers as effectively as possible. So here are my answers to the following ...
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21 views

How to calculate normal (of magnitude 1) of a triangle?

I am currently doing a bit of geometry practice and wanted to know how to calculate the normal (of magnitude 1) of a triangle defined by 3 vertices: a, b and c`. ...
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0answers
30 views

three dimensional subspace question

If a vector is in $\mathbb{R}^5$, does this mean that the projection of this vector onto $S$ is in $\mathbb{R}^3$, where $S$ is some 3-dim subspace of $\mathbb{R}^5$?
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Higher Dimensional Right-Hand Rule

In seven dimensions, the cross product makes sense. Without resorting to nonvector tensors or exterior products (although they can be used to further explain), how does one perform this cross product ...
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3answers
39 views

Is it possible for a set of non spanning vectors to be independent?

I was reading about linear spans on Wikipedia and they gave examples of spanning sets of vectors that were both independent and dependent. They also gave examples of non spanning sets of vectors that ...
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1answer
24 views

Inner product over the $C^2$

Let a, b, c, d ∈ C and consider the vector space $C^2$ Suppose inner product is defined as: $⟨x, y⟩ = ax_1\bar y_1 + bx_2\bar y_1 + cx_1\bar y_2 + dx_2\bar y_2$ I am trying to find all a, b, ...
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1answer
14 views

is the space of finite sequences Frechet?

Let $E:=\coprod_{\mathbb{N}}\mathbb{R}$ be the space of all finite, real sequences equipped with the final structure wrt to all injections $inj_k(x)=(0,0,\dots,x,0,\dots)$. Since completeness of $E$ ...
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0answers
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Vector Calculus Identities - Suffix Notation

Using Cartesian coordinates, show that $$ (u \cdot \nabla)u = \frac{1}{2}\nabla (u \cdot u) - u \wedge(\nabla \wedge u), $$ and hence that: $$ \nabla \wedge (u \cdot \nabla)u = (\nabla \cdot ...
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1answer
40 views

Find a basis of the $k$ vector space $k(x)$

Suppose $x$ is a transcendental over field $k$ and $k(x)$ is the field of fractions of $k[x]$. Can we explicitly express a basis of the $k$ vector space $k(x)$?
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1answer
30 views

Vectorspace subspace proof

V is $n$ dimensional vectorspace over $\mathbb{R}$. $W\subset V$ is $m$ dimensional subspace over $\mathbb{R}$ and $m < n$. $$Y=\bigcap \{U:U \text{ is a subspace of} \ V, \dim U=n-1, W \subset U ...
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2answers
50 views

$l^r \subset l^p$ and is it even a subspace

It is true that for $r<p$ and $r,p \in [1,\infty)$ we have that $l^r \subset l^p$. Is it true that $l^r$ cannot be isomorphic to a subspace of $l^p$?
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General Vector Space: Change of basis

If $P$ is the transition matrix from a basis $B'$ to a basis $B$, and $Q$ is the transition matrix from $B$ to a basis $C$, what is the transition matrix from $B'$ to $C$? What is the transition ...
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1answer
28 views

Proving linear dependence

$V=\mathbb{R}^{\mathbb{R}}$ vectorspace and $f_1,f_2\in V$, so that : $$\forall c_1,c_2 \in \mathbb{R} \ (\forall x\in \mathbb{R} \ c_1f_1(x)+c_2f_2(x)\geqslant 0 \ \vee \ \forall x\in \mathbb{R} \ ...
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2answers
22 views

How to find a normal vector from an equation in the form f(x,y)?

If I have an equation $f(x,y)$ which given the $x$ and $y$ coordinate, it gives you the $z$ coordinate. How can I find the normal (directional) vector of the the point $(x,y,f(x,y))$? This would be ...
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1answer
179 views

Function psi, vector potential, satisfying conditions

Using spherical polar coordinates ($r, \theta, \phi$) verify that the vector $F = r^{-2}e_r$ is solenoidal. Find the function $\psi(r, \theta)$ such that $A = \frac{\psi(r, \theta)}{rsin ...
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1answer
40 views

find the number of solutions of the equation $a_1x_1+a_2x_2+…+a_nx_n=0$ in a linear space over Galois field

Linear space $\Bbb F_p^n$ contains $p^n$ vectors $( x_1, x_2, ..., x_n)$ with length $n$ over finite $\Bbb F_p$ Galois field comprised from $p$ elements. How many solutions in $\Bbb F_p^n$ has the ...
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19 views

Confusion with proving that some subspace of a Banach-Space is closed

So here is my problem, I am trying to show that, Let $X,Y$ be Banach-Spaces and $T:X\rightarrow Y$ a linear and bounded map. Then $T(X)$ is closed if $Y/T(X)$ is of finite dimension. While ...
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29 views

Finding a basis for span of vectors

U = span{(1,0,0),(0,2,-1)}. W = span{(0,1,-1)} How can I find bases for U and W ? (I think they're linearly independent, right?) can I just take B1 = {(1,0,0),(0,2,-1)} for U, and B2 = {(0,1,-1)} for ...