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 we ...
1
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3answers
68 views
Revisited: $[T]^{\gamma}_{\beta}$ is diagonal?
Let $V$ and $W$ be finite-dimensional vector spaces with $\dim(V)=\dim(W)$ and let $T:V \rightarrow W$ be a linear map. How do I prove that there are bases $\beta$ of $V$ and $\gamma$ of $W$ such that ...
-2
votes
2answers
37 views
$T:V\rightarrow W$ such that $R(T) \subset W'$ is a subspace of ${\cal{L}}(V,W)$ [closed]
Let $V$ and $W$ be finite-dimensional vector spaces over $F$ and $W'\subset W$ a subspace, then the subset ${\cal{L}}(V,W)$ consisting of all linear maps $T:V\rightarrow W$ such that $R(T) \subset W'$ ...
0
votes
1answer
45 views
Existence Proof: $T(v_i)=w_i$ for all $i=1,2,3,\dots,n$
Theorem to prove:
Let $\{v_1,\dots,v_n\}$ be a linearly independent set in a finite-dimensional vector space $V$ and let $w_1,\dots,w_n$ be arbitrary vectors in a vector space $W$. Then there exists ...
1
vote
1answer
28 views
Collinearity of three points of vectors
Show that the three vectors $$A\_ = 2i + j - 3k , B\_ = i - 4k , C\_ = 4i + 3j -k$$ are linearly dependent. Determine a relation between them and hence show that the terminal points are collinear.
...
3
votes
2answers
54 views
Proof of the linear independence of the generalized eigenvectors of a square matrix
I'm currently stuck on this problem:
Let $V$ be a finite dimensional vector space. If $S: V\rightarrow V$ and $T: V\rightarrow V$ are linear maps and $ST=TS$, prove every eigenvalue of $ST$ is a ...
1
vote
2answers
54 views
What is the role of supremum in operator norm
An operator norm is defined as $\|A\|_S=\sup\{\|Av\|:v\in \Bbb R^n, \|v\|=1\}$. Where $\|\cdot\|$ is some norm on $\Bbb R^n$ and $A\in M_n(\Bbb F)$, space of square matrices of dimension $n$ over ...
0
votes
1answer
45 views
$T:P_4(\mathbb{R})\rightarrow P_4(\mathbb{R})$ such that $N(T) = P_1(\mathbb{R})$ and $R(T)=P_2(\mathbb{R})$
So, I'm asked to give an example of a linear map $T:P_4(\mathbb{R})\rightarrow P_4(\mathbb{R})$ such that $N(T) = P_1(\mathbb{R})$ and $R(T)=P_2(\mathbb{R})$.
As far as I understand, I'm trying to ...
0
votes
1answer
35 views
How to write this proof formally?
I have to prove that if $V$ is an unitary vector space and $W,U$ are subspaces of $V$, then $W^\bot \cap U^\bot=(W\land U)^\bot$ where $\bot$ means orthogonal complement and $\land$ is conjunction of ...
0
votes
1answer
37 views
$[T]^{\gamma}_{\beta}=[v]_{\gamma}$ with $\beta=\{1\}$ a basis for $F$
Let $V$ be a finite-dimensional vector space over $F$ with basis $\gamma$ and let $v\in V$. Find a linear map $T:F \rightarrow V$ such that $[T]^{\gamma}_{\beta}=[v]_{\gamma}$, where $\beta = \{1\}$ ...
0
votes
1answer
88 views
How Does One Find A Basis For The Orthogonal Complement of W given W?
I've been doing some work in Linear Algebra for my course at school. I just want to be clear about how to find the orthogonal complement of a subspace. The basis for the subspace, W, is shown below, ...
4
votes
4answers
95 views
$T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$
How should one prove that there exists a linear map $T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$ if $\dim(V')+\dim(W')=\dim(V)$, where $V$ and $W$ are finite-dimensional ...
0
votes
1answer
23 views
Equal Shape: Recovering an Isomorphism Between $M_{3\times 2}(F)$ and $P_5(F)$
I'm asked to find an isomorphism between $M_{3\times 2}(F)$ and $P_5(F)$, but what does it mean for a $3\times 2$ matrix to have an inverse?
0
votes
2answers
42 views
Can isomorphism existence be proved via rank-nullity?
A student in my class mentioned his approach to proving isomorphism via the rank-nullity theorem rather than showing an inverse's existence. I didn't quite understand how, but perhaps someone here ...
2
votes
3answers
74 views
Revisited: How is $\phi:{\cal{L}}(V,W)\rightarrow M_{m\times n}(F)$ an isomophism of vector spaces?
I'm told in lecture that if $V,W$ are vector spaces over $F$ and ${\cal{L}}(V,W)$ is the vector space of all linear maps $V\rightarrow W$ and ${\scr{B}}$ and ${\scr{C}}$ are bases for $V$ and $W$ ...
0
votes
2answers
46 views
If the null space of a $8 \times 6$ matrix is $1$-dimensional, what is the dimension of the row space?
I understand how to find the row space, column space, and At space given the rank of the matrix, but how would I find this? Does the dimension just mean the rank? Would the row space be $5$?
Can you ...
0
votes
1answer
29 views
3 equations with 9 unknown variables with scalar product
Excuse my bad english pls. I can't find a proper solution to my problem because i don't know the exact mathematical terms in english.
My problem is how to get the 3 elements of each of 3 vectors ...
0
votes
1answer
149 views
Subspaces, transformation matrices exercise
I have trouble understanding the following exercise so I would really appreciate any help you could give me:
Let $k$ be a non zero vector in $\mathbb R^n$, written in standard basis. Here is a ...
0
votes
2answers
50 views
What is $[T]^{\scr{C}}_{\scr{B}}$?
What does it mean for $[T]^{\scr{C}}_{\scr{B}}\in M_{m\times n}(F)$ to be a matrix of $T$ in basis $\scr{B}$ in $\scr{C}$?
3
votes
1answer
22 views
Notating each element of a vector which already has a subscript
If I had a vector $\mathbf{x}$, I would denote element $i$ as $x_i$.
However, if my vector already has a subscript, for example $\mathbf{x}_j$ or $\mathbf{x}_{10}$, how should I show element $i$?
I ...
2
votes
1answer
56 views
Scalar product with ON-base $e_1,e_2,e_3$
Get the vector u which length is 4, in the ON-base $$e_1,e_2,e_3$$ and the baseangles $$\frac{\pi}{3}, \frac{5\pi}{6}, \frac{\pi}{2}$$
1
vote
2answers
58 views
How do I determine if given set is a subspace of the specified vector space (answer provided)?
Determine (with proof) if given set is a subspace of the specified vector space.
The set of vectors $$ \begin{pmatrix} a+3b-c \\ 2b-4c \\ 5a + 6c\\ 0 \end{pmatrix}$$ in $\Bbb R^4$
My answer is:
...
0
votes
0answers
52 views
Showing that the properties of dot product hold for the trace of a matrix
This is the question:
"Consider the vector space of 2x2 matrices. For a matrix A, we de fine the trace of this matrix
to be $Tr(A) = A_{11} + A_{22},$ that is, the sum of the diagonal components.
(a) ...
0
votes
0answers
25 views
Find a matrix $P$ such that $[U]_B = P[U]_A$ for all upper triangular matrices $U$
This is the homework problem:
"Consider the following two basis sets for the vector space of all (2x2) upper triangular matrices
U:
$$
A= \begin{Bmatrix} \begin{bmatrix}
1 & 1 \\
...
1
vote
0answers
28 views
It is not possible to introduce multiplication in $v_n$(For $n>2$) so as to satisfy all field properties
In the book Calculus Vol 1- Tom M. Apostol .Before beginning to define the dot product of two vectors he tells
It can be shown that except $n=1, 2$, it is not possible to introduce multiplication ...
1
vote
2answers
39 views
Linear functional $\mathscr{L}(E,F)$
Let $\mathscr{L}(E,F)$ denote the space of all linear functionals from $E \to F$.
Let $\mathscr{C}(E,F)$ denote the space of continuous linear functionals from $E \to F$. My question:
How to prove ...
0
votes
1answer
25 views
Linear Algebra Proof of Injective Function
I'm new in the University and I don't know how to solve this:
Suppose $v$ is a non null element of a vector space $V$ on $\mathbb R$. Show that the function is injection:
$\mathbb R\to V $
$t ...
1
vote
2answers
46 views
Need help understanding the concept of the Jacobian Matrix and its relation to differentiation
As the question suggests, I need help understanding the concepts around the following differentiation of vector-valued functions:
1) The Norm. I understand that Norm to be defined as follows:
In the ...
1
vote
1answer
61 views
Calculating mean velocity of an orbiting body as it moves towards a point.
I'm making a game, in the game planets orbit a central point in circular orbits, they move directly towards their targets and the vector is simply added to their orbital path. Whilst not realistic it ...
-4
votes
0answers
126 views
$[T]_{\scr{C}}=[T]_{\scr{B}}^{\scr{C}}[v]_{\scr{B}}$: A Desideratum for Creative Disquisition [closed]
$\blacklozenge\hspace{.2cm}$Consider the following data in this first litany:
$\hspace{2cm}\Diamond\hspace{.5cm}$$V$ and $W$ are finite-dimensional vector spaces.
...
0
votes
2answers
52 views
Find a perpendicular line to another line inside a given plane
I have the line
$$D: \frac{x-7}{12} = \frac{y + 1}{-6} = \frac{z-2}{-2}$$
that crosses the plane
$$P : -4x - 5y - z -3 = 0$$
on point $A=(47/4, -67/8, 41/8)$.
I must find a line from point $A$ that ...
0
votes
1answer
43 views
Equation of a plane from 2 lines
I have two lines with the following equation
$$D1 : (x, y, z) = (2,0,0) + k(0,3,0)$$
$$D2 : (x, y, z) = (2,0,2) + k(0,0,1)$$
and I must find out the equation of the plane that they make. I ...
0
votes
1answer
89 views
Verify $X$ is a direct sum of $V$ and $W$
Let
$$\begin{align*}
X&=\mathcal{C}[0,1],\\
V&=\{v\in \mathcal{C}[0,1]\mid v(x)=v(-x)\},\\
W&=\{w\in\mathcal{C}[0,1]\mid w(x)=-w(-x)\}.
\end{align*}$$
Is it possible to verify that $X$ is ...
0
votes
1answer
12 views
Determine when 2 mobile will cross when traveling on the same direction
I have the following problem
Two mobiles are traveling in the same direction on the line with this equation
(x + 3) / 10 = (y + 10) / 20 = (z -10) / -20
The ...
1
vote
2answers
101 views
$\infty$ | Span | Basis
Let $V$ be a finite dimensional vector space and $S \subset V$ a subset (possibly
infinite) with $\operatorname{span}(S) = V$. Does there exist a subset of $S$ that is a basis for $V$?
3
votes
5answers
220 views
Dimensions: $\bigcap^{k}_{i=1}V_i \neq \{0\}$
Let $V$ be a vector space of dimension $n$ and let $V_1,V_2,\ldots,V_k \subset V$ be subspaces of
$V$. Assume that
\begin{eqnarray}
\sum^{k}_{i=1} \dim(V_i) > n(k-1).
\end{eqnarray}
To show that ...
1
vote
0answers
23 views
Reverse of rotate a vector around an axis - Finding North with magnometer
I've been reading the wikipedia page on rotation matrices and I know the reverse (or at least a very related) version of this question has been asked many times before on this site. However my ...
-4
votes
0answers
96 views
Properties of Subspace Sums [closed]
Read this comment
"If it is not too taxing, perhaps you could wind out what you've propounded in terms more native to the simple, elementary content of an honors linear algebra course."
in ...
0
votes
1answer
30 views
Codimensionality: On Cardinality of Linear Equations
How does the codimension of a subspace give the number of linear equations needed to define the subspace?
2
votes
1answer
29 views
number of elements in vector space
Given that $k$ is a finite field with $q$ elements and $V$ is a $n$-dimensional $k$-vector space, then by basis representation, we know that for $v \in V, v=a_1v_1+a_2v_2+\cdots+a_nv_n$ uniquely. ...
2
votes
1answer
57 views
Convex cone question.
Hoi, let $V$ be finite dimensional real vector space with inner product $\left\langle . \right\rangle$ and let $\Gamma \neq \{0\}$ be a closed convex cone. Let $$\Gamma_0^{\perp}:=\{v\in ...
0
votes
2answers
22 views
Magnitude of Axis vectors in question
I have a question on my revision sheet:
Write the vector, v=-2 i + 4 j , in polar form.
is it safe to assume axis vectors i and j have a magnitude of 1?
2
votes
2answers
26 views
Function space of a finite set and $\Bbb R^n$
I read in a tutorial that a function space $F(S, \mathbb{R})$ of a finite set $S$ of cardinality $n$ has dimension $n$. To be clear $F(S, \mathbb{R})$ is the set of all functions defined on the set ...
1
vote
1answer
209 views
Angle between functions
I have a rather simple question but googling it did not bring a satisfactory result:
Assume you have given two function $f$ and $g$ on some space $\mathcal{L}^2(\Omega)$ where $\Omega \subset ...
1
vote
2answers
58 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.
1
vote
2answers
37 views
Dimension Recovery of $S \subset P_n(F)$
How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that
$f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset?
-2
votes
4answers
100 views
Dimensionality and Subspace Existence: A Potential Outlet for Disquisition
The subset of $F^n$ consisting of all vectors $(a_1,a_2,\dots,a_n)$ such that $a_1+a_2+\cdots+a_n=0$ is a subspace of $F^n$ and its dimension is ...(?)....
Initially, my intuition said the ...
1
vote
1answer
63 views
Kenneth Hoffman | Ray Kunze: An Inquiry into Symbolic Meaning
When those authors state the following
$\bf{Theorem 6.}$ If $W_1$ and $W_2$ are finite-dimensional subspaces of a vector space $V$, then $W_1+W_2$ is finite-dimensional and
\begin{eqnarray}
\dim ...
0
votes
2answers
38 views
What does it mean that a finite set in a vector space has this property?
My homework problem says to let $S$ be a finite set in a vector space $V$ with the property that every $\vec x$ in $V$ has a unique representation as a linear combination of elements of $S$. Show that ...
1
vote
1answer
34 views
On the Dimensionality of Space: An Elementary Analysis
The below theorem I am to prove. Perhaps you have a critique...
Theorem 2.4 Let $W_1$ and $W_2$ be two subspaces of a vector space $V$. Then $\dim(W_1 \cap W_2)=\dim(W_1)$ if and only if $W_1 ...
2
votes
1answer
54 views
Proving/disproving this is a linear subspace
I need to prove/disprove that $W$ is a linear subspace, and I'm not sure my approach is correct (especially the last point I'm making). Please correct me if I'm wrong.
Let $V$ be a set of vectors ...
