1
vote
0answers
18 views

finding the symmetric point

let there be $4$ points. $A(-1,1,1), B(2,0,-1), C(1,3,-2), D(-2,-1,0)$. the $4$ points are not on the same line. the plane which goes through the points $A$ and $B$, and which is also paralel to the ...
1
vote
0answers
35 views

having trouble with a 3-dimensional basis-change problem/

Let $V$ be a 3d vector space with a chosen basis $\alpha=\{e_1,e_2,e_3\}, \beta=\{f_1,f_2,f_3\}$ for $V$ satisfying: $$\begin{align}e_1 & =f_1+f_2+f_3 \\ e_2 &=f_2+2f_3 \\ e_3 & =f_3 ...
0
votes
2answers
37 views

i am having trouble with one of the homework question regarding to linear algebra(vector and span)

$V$ is a vector space of some dimension, with $\vec u,\vec v,\vec w$ independent set of vectors in $V$. define the subspace of $V$ given by $W = \operatorname{span}(\vec u-\vec v+\vec w, 2\vec u+\vec ...
1
vote
1answer
27 views

Direct Sum of Three Subspaces

Suppose $U = \{(x, y, x+y, x -y, 2x) \in \Bbb F^5 : x, y \in \Bbb F\}$. Find three subspaces $W_1, W_2, W_3$ of $\Bbb F^5$, none of which equal $\{0\}$ such that $\Bbb F^5 = U \oplus W_1 \oplus W_2 ...
1
vote
1answer
10 views

Finding the conditions of (x,y,z,t) for them to belong to the span of a set of vectors

So I got this math exercise, and I don't know how to go about it: In $\mathbb{R}^4$, $S$ is the subspace spanned by the following set of vectors: $(1, 1, 1, 0) , (1, 2, 1, 1) , (2, 0, 1, 1) , (3, 0, ...
1
vote
1answer
38 views

Coordinate matrices in standard basis

If $A=1+2x+4x^3$ and $B=2+3x^2+x^3$ are vectors in Polynomial space, find out the coordinate matrices for $A$ and $B$ in standard basis and hence find out the angle between vectors A and B.
0
votes
2answers
30 views

vector question assistance

let there be 2 lines: $(2,-3,1) + s(3,-2,1)$ and $(2,-1,-3) +t(3,-2,1)$ which are parallel to each other. find the formula of the plane determined by them. my try: a vector perpendicular to ...
2
votes
1answer
44 views

Vector calculation question

the points a b c d are concordantly ( 1,2,-3) , (-1,2,1) , ( 0,1,-2) , ( 2,-1,1) find formula of the plane going thorugh d and which is pararlel to plane abc calculate the volume of pyramid abcd. ...
0
votes
0answers
21 views

Vector pyramid question

Suppose we have a pyramid with two vectors known, as well as the angle between them...if we mutpliy their size by each other mutiply by the sine of the said angle and then by sixth, will we get the ...
1
vote
2answers
48 views

Weird field notation

I have a question: Let $\mathbb{F}$ be any field characteristic $0$. Recall that $x_i$, denotes the $i^{th}$ entry of a vector $x\in\mathbb{F}^n$. Define $$S = \{x\in\mathbb{F}^5 \mid x_i = ...
3
votes
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}
0
votes
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 ...
0
votes
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 ...
2
votes
2answers
57 views

When $ax+by+cz+d=0$ is a plane, $a^2 + b^2 + c^2 \neq 0$

I'm reading a book about equation of planes and an way to determinate the equation is to suppose a point $P = (x, y, z)$ And suppose also that $A=(x_0, y_0, z_0)$ is in the plane. $P$ is in the plane ...
0
votes
0answers
21 views

Show that $\{w^{1/2}\phi_n\}$ is an orthonormal set in $L^2(D)$ if $\{\phi_n\}$ is an orthonormal set in $L^2_w(D)$

As mentioned in the title, my problem is: Show that $\{w^{1/2}\phi_n\}$ is an orthonormal set in $L^2(D)$ if $\{\phi_n\}$ is an orthonormal set in $L^2_w(D).$ So I know that: ...
3
votes
1answer
54 views

Vector spaces - $\min\{p\in\mathbb{N}|\text{ker}f^p=\text{ker}f^{p+1}\}=\min\{q\in\mathbb{N}|\text{im}f^q=\text{im}f^{q+1}\}$

$E$ is a $\mathbb{K}$ vector space, $f\in\mathcal{L}_\mathbb{K}(E)$. Let $p\in\mathbb{N}$ so that $\text{ker} f^p=\text{ker}f^{p+1}$ and $q\in\mathbb{N}$ so that $\text{im} f^q=\text{im}f^{q+1}$ ...
1
vote
1answer
33 views

Proof that Vector Space in Domain of Linear Map is a Direct Sum

I'm working through problems in Linear Algebra just for fun and I am getting stuck on Axler 3.4. Suppose that $T$ is a linear map from $V$ to $\mathbf{F}$. Prove that if $u \in V$ is not in $null\ ...
1
vote
1answer
49 views

Is there a specific method to finding a basis for vector spaces over $\mathbb{Q}$ ?

I am stuck on the first one but there are 5 questions on this so I really need help with the process. If anyone can help with any of the following. i) Find a Basis for the field K = ...
0
votes
2answers
37 views

Prove that the linear space of polynomials with root $\alpha \in \mathbb{R}$ is a subspace of $\mathbb{R}[x]_n$

Prove that linear space of polynomials having root $\alpha \in \mathbb{R}$ is a subspace of $\mathbb{R}[x]_n$. It's also required to find basis and dim of that subspace. I recently started learning ...
0
votes
1answer
31 views

Show that $f(x)$ is orthogonal to $f'(x)$ in $L^2(-\pi, \pi)$

I have the following problem: Suppose $f$ is of class $C^{(1)}$, $\;2\pi$-periodic, and real-valued. Show that $f'$is orthogonal to $f$ in $L^2(-\pi, \pi)$ by a) expanding $f$ in ...
0
votes
2answers
42 views

Why are these vectors expressed as row vectors and not column vectors? When to write as row vectors or column vectors?

Everytime I have been asked to find a basis when the vectors were given in comma delimited form, I, and the book, would write out the vectors as columns in a matrix. Another example in the book ...
0
votes
1answer
35 views

Sketching a vector of a cyclist's route

A cyclist travels at a steady $16 km/h$ on the four legs of his journey. From the origin O he first travels NE (north-east in the direction of the vector i + j) for one hour to the point A. He then ...
1
vote
2answers
24 views

Prove that $\lim_{t\to t_0}[f(t) \times g(t)]=u \times v$

Let $f(t)=(f_1(t),f_2(t),f_3(t))$, $g(t)=(g_1(t),g_2(t),g_3(t)).$ $$\lim_{t\to t_0}f(t)=u; \lim_{t\to t_0}g(t)=v.$$ Prove: $$\lim_{t\to t_0}[f(t) \times g(t)]=u \times v$$. Thanks ahead:)
0
votes
3answers
37 views

Vectors and orthonormal basis vectors help!

I'm not entirely sure how to go about answering this question about vectors. Any advice/help is appreciated. Write the vector $\displaystyle a =\begin{bmatrix}3\\-1\\7\end{bmatrix}$ as a linear ...
0
votes
2answers
63 views

Find the resulting speed and direction. Trig Problem involving resultant and vectors.

A barge is pulled by two tugboats. The first tugboat is traveling at a speed of 15 knots with heading 130°, and the second tugboat is traveling at a speed of 11 knots with heading 190°. Find the ...
0
votes
1answer
63 views

Problem involving Bearing, Heading, and True Course.

A plane is flying with an airspeed of 170 miles per hour and heading 150°. The wind currents are running at 30 miles per hour at 170° clockwise from due north. Use vectors to find the true course and ...
3
votes
3answers
83 views

Prove that $\mathbb{Z}$ is not isomorphic to additive group of any vector space over any field.

Prove that $\mathbb{Z}$ is not isomorphic to additive group of any vector space over any field. Proof. Surpose that: $\phi : (A, +) \rightarrow \mathbb{Z} $ is an isomorphism. Then there is some ...
1
vote
2answers
52 views

Proof: $\exists$ subspace $U$ of $ker(f)$ with $U \bigoplus T_1 = T_2 $

I need help with this proof: Let $V, W$ be K-vectorspaces. Let $T_1, T_2$ be subspaces of V with $T_1 \subseteq T_2$. Let $f \in hom_K(V,W)$. Show the following: If $ f(T_1) = f(T_2)$ then exists a ...
0
votes
1answer
61 views

Invariant subspaces for endomorphisms with associated Jordan matrices

I would like to know which are the invariant subspaces for the endomorphisms $f1$, $f2$, $f3$, $f4$, $f5$ from vector space $V$ that have the next associated Jordan matrices: $J1 = \left( ...
0
votes
2answers
24 views

How to prove perpendicular vectors problem

If I have: $$|\underline a + \underline b| = |\underline a - \underline b|$$ how do I prove that $\underline a$ is perpendicular to $\underline b$?
0
votes
1answer
28 views

Vectors In Three Dimensions

Hi! I am working on some online homework for my calc2 class and I am having trouble with this problem. I first set $r_1$ and $r_2$ equal to one another to get $(-1-4t, 2+2t, -14+2t)=(-13+4t, 8-2t, ...
0
votes
1answer
63 views

Banach's Fixed Point theorem - banach vector space

Problem 1 (Banach fixed point theorem). Let $(V, || \, \, \, ||)$ be a Banach space, $U \subset V$ a closed subset (in the sense that convergent sequences in $U$ have their limits in $U$) and $T : ...
1
vote
2answers
96 views

Linear Algebra, Vector Space: how to find intersection of two subspaces ?

$${ W = Sp\{{(1,3,4),(2,5,1)\}}\\ U = Sp\{{(1,1,2),(2,2,1)}} \}$$ Find a span $${U \bigcap W}$$ First time using Math latex, pretty hard.
0
votes
1answer
48 views

finite and infinite, vector space, linear transformation

I have to answer these questions for homework and I don't know if I'm answering these correctly. I think most of them are correct, but a double check would be much appreciated. a) If $S$ is a set ...
2
votes
1answer
50 views

Prove that if $v_1,v_2,…,v_r$ form a linearly independent set of vectors in $V$…

Let $S$ be a basis for an n-dimensional vector space $V$. Prove that if $v_1,v_2,...,v_r$ form a linearly independent set of vectors in $V$, then the coordinate vectors $(v_1)_S,(v_2)_S,...,(v_r)_S ...
0
votes
2answers
50 views

Proof that if we add a vector to a linearly dependent set of vectors in a vector space $V$, then the new set of vectors is still linearly dependent

Prove that if $S=\{v_1, v_2, v_3\}$ is a linearly dependent set of vectors in a vector space $V$, and $v_4$ is any vector in $V$ that is not in $S$, then $\{v_1, v_2, v_3, v_4\}$ is also linearly ...
2
votes
3answers
76 views

Find dimension of ℒ $(V)$ and polynomial that brings every linear transformation to $0$

Here's the prompt: Let V be a vector space of finite dimensions $n$ over the field $\mathbb{F}$, and let $\tau \in$ ℒ $(V)$. What is the dimension of ℒ $(V)$ as a vector space over $\mathbb{F}$? With ...
0
votes
0answers
28 views

Why is $\hat{x}$ in the linear regression equation $A^TA\hat{x} = A^Tb$ part of $C(A^T)$

When finding the best fit line for a number of points, we use $A^TA\hat{x} = A^Tb$ where we solve for $\hat{x}$. I understand that the projection $p=A\hat{x}$ is part of the column-space of $A$ and ...
1
vote
3answers
88 views

An infinite generating set of a finite dimensional vector space contains a basis

Let $S$ be an infinite generating set of a finite dimensional vector space , then how do we prove that there is a subset of $S$ which is a basis of the vector space ? Please help
0
votes
1answer
64 views

Question on sequence space (as a linear space)

Let $X$ be the space $\ell_\infty$ of all bounded sequences of real scalars. If $Y$ is the set of all $x\in X$ that have bounded partial sums (1) Can I say $Y$ is a linear space (as a subspace of ...
0
votes
1answer
28 views

Vectors and Planes

Let there be 2 planes: $x-y+z=2, 2x-y-z=1$ Find the equation of the line of the intersection of the two planes, as well as that of another plane which goes through that line. Attempt to solve: the ...
0
votes
0answers
8 views

Geometry of Vectors question

Let there be 3 points; $A(1,2,0), B(2,2,-1), C(4,0,1)$. Find the plane in which all $3$ lie, and find point $D$ such that $ABCD$ is a parallelogram . I did find the plane equation which is ...
0
votes
0answers
26 views

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:$$ ...
2
votes
2answers
13 views

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 ...
0
votes
1answer
28 views

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 ...
0
votes
2answers
41 views

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 ...
0
votes
2answers
51 views

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 ...
1
vote
1answer
20 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 ...
0
votes
1answer
47 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 ...
1
vote
1answer
28 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 ...