Tagged Questions
0
votes
0answers
20 views
What does this mean: Symmetry of the KDV generated by a vector field
What is
a symmetry of the KDV
$$\frac{\partial u}{\partial t}=6u\frac{\partial u}{\partial x}-\frac{\partial^3 u}{\partial x^3}$$ generated by $$V=A(t,x,u)\frac{\partial }{\partial ...
1
vote
1answer
28 views
Find vectors vertical to given vectors with certain length
Given the vectors $\mathbf{u,v}$ in R³, determine all vectors that are
vertical to $\mathbf{u}$ and $\mathbf{v}$ with length = 1
Every vector $\mathbf{x'}$ that is to be found must meet these ...
0
votes
1answer
18 views
Vector Field, Scalar Field. Which is meaningful and not?
Let a, b, c be vectors, f(x, y, z) be a scalar field, F(x,y,z) be a vector field. Which of the
following expressions are meaningful?
I. (a×b)×(c×b)
II. |a|(b· c) +|a|(b+c)
III. ∇ ×(f F)
IV. (∇ ...
1
vote
3answers
38 views
Calculate two vectors given their norms and angle
For two vectors $\mathbf{u,v}$ in $\mathbb{R}^n$ euclidean space, given:
$\|\mathbf{u}\| = 3$
$\|\mathbf{v}\| = 5$
$\angle (\mathbf{u,v})=\frac{2\pi}{3}$
Calculate the length of ...
0
votes
1answer
21 views
Linear interpolation of points in isometric isomorphic spaces
Suppose that we have two spaces $\mathcal{F}$ and $\mathcal{H}$ and we know that $\mathcal{H}$ is isometric isomorphic to $\mathcal{F}$, so that distances and angles are preserved. Note that we are ...
1
vote
2answers
22 views
Prove using an example that there is no plane on R3 that contains every group of 4 points
Well, this is a homewrok question (which I know I should not be asking, but I cannot find an answer to this anywhere):
The exercise is as follows:
i) Find the equation of the plane of R3 that ...
1
vote
0answers
34 views
Proving that the circumcenter is the centroid
Given a triangle and its centroid, we know that the 3 line segments between the centroid and each of the vertices of the triangle divide the triangle into three smaller triangles. Prove that the ...
1
vote
2answers
32 views
Is the matrix V in the subspace U?
I'm given that $U$ is the subspace of $M(3,2)$ generated by
$A=\begin{bmatrix} 0 & 0 \\ 1 & 1 \\ 0 & 0 \end{bmatrix}$,
$B=\begin{bmatrix} 0 & 1 \\ 0 & -1 \\ 1 & 0 ...
0
votes
1answer
61 views
Examples on the dimension of vector spaces of real functions
Let $S$ be a vector space of functions from $\mathbb{R}^n$ to $\mathbb{R}$, say $S := \{ f:\mathbb{R}^n \rightarrow \mathbb{R} \}$.
I am looking for some examples in which the dimension of $S$ is ...
0
votes
1answer
47 views
Prove that in a vector space $V$ over field $\mathbb{F}$ $0\cdot v=0$
Prove that in a vector space $V$ over field $\mathbb{F}\space$:
$0\cdot v=0$ for all $v \in V$
I started by proving that for all $x \in F$
$x \cdot 0 = 0$ using the Field axioms.
Then I said that ...
2
votes
2answers
34 views
Picard iterations of a matrix
I need help with this problem. I think i got the first three questions of the exercise, but i'm stuck at the fourth one.
We consider the map $T:{\mathbb{R}^2}\longrightarrow{\mathbb{R}^2}$ defined ...
0
votes
2answers
50 views
Volume of a parallelepiped, given three vectors
I want the volume of a parallelpiped and I have the three vectors $$4e_1+2e_2-e_3$$$$e_1-3e_2-2e_3$$$$2e_1-e_2+3e_3$$ that coinciding with three of the parallelpipeds sides. HON-base
I made it into a ...
0
votes
0answers
48 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:
$$
A= \begin{Bmatrix} \begin{bmatrix}
1 & 1 \\
...
2
votes
1answer
56 views
Scalar product with ON-base $e_1,e_2,e_3$
Get the vector u which length is 4, in the ON-base $$e_1,e_2,e_3$$ and the baseangles $$\frac{\pi}{3}, \frac{5\pi}{6}, \frac{\pi}{2}$$
0
votes
2answers
36 views
What does it mean that a finite set in a vector space has this property?
My homework problem says to let $S$ be a finite set in a vector space $V$ with the property that every $\vec x$ in $V$ has a unique representation as a linear combination of elements of $S$. Show that ...
0
votes
2answers
53 views
Is vector subtraction commutative?
Is Vector Subtraction commutative (a-b = b-a)? And if so how is it visually represented?
My textbook states that it is, but I can't seem to figure out how to visually represent it with the ...
2
votes
1answer
53 views
Proving/disproving this is a linear subspace
I need to prove/disprove that $W$ is a linear subspace, and I'm not sure my approach is correct (especially the last point I'm making). Please correct me if I'm wrong.
Let $V$ be a set of vectors ...
0
votes
4answers
65 views
Given two column vectors $a$ and $b$, what is the determinant of $A$ if $A=ab^T$
Given two column vectors $a$ and $b$ in $\mathbb R^n$
, $n \ge 2$, form the $n×n$ matrix
$A = ab^T$. What is the determinant of $A$?
(Hint: Examine linear
dependence).
2
votes
1answer
49 views
Proving linear independence
Let $A$ be an $n \times n$ matrix and suppose $v_1, v_2, v_3 \in \mathbb{R}^n$ are nonzero vectors that satisfy:
$$
Av_1 = v_1 \\
Av_2 = 2v_2 \\
Av_3 = 3v_3 $$
Prove that $\{v_1, v_2, v_3\}$ is ...
2
votes
3answers
38 views
Linear algebra - how to tell where vectors lie?
I'm working my way (self-study) through Strang's text on Linear Algebra and am currently on Problem 1.2 #6.
6b) The vectors that are perpendicular to $V = (1,1,1)$ lie on a _ .
6c) The vectors that ...
0
votes
1answer
41 views
Kernels of Adjoints
Let $A$ be an $m \times n$ matrix. Show that $\mbox{Ker} A = \mbox{Ker} (A^*A)$.
To do that you need to prove 2 inclusions, $\mbox{Ker} (A^*A)$ is a subset of $\mbox{Ker} A$ and $\mbox{Ker} A$ is a ...
2
votes
2answers
37 views
Generalised eigenvalue is eigenvalue if it is in the field
I would like to prove the following assertion:
Let $\mathscr{F}$ be a field and $\mathscr{\phi}$ be an $\mathscr{F}$-linear endomorphism of a finite dimensional $\mathscr{F}$-vector space ...
2
votes
1answer
59 views
Vector Spaces Linear algebra
bI've been working through some problems in my Linear Algebra course and I've come across some that have me confused. I'm not particularly good at vector spaces so some help would be greatly ...
1
vote
1answer
36 views
Question on finding an orthogonal complement
So I have a practice question, and I want to make sure that my understanding of the concept is valid. This is the text of the question:
Find the orthogonal complement of the subspace of $ R^3$ ...
2
votes
1answer
25 views
Finding the basis for a plane
I know the conditions of being a basis. The vectors in set should be linearly independent and they should span the vector space.
However, how do I find two different sets of basis vectors of the ...
1
vote
1answer
33 views
linear combination
If $\mathbb C$ is the field of complex numbers which vectors in $\mathbb C^3$ are linear combinations of $(1,0,-1)$,$(0,1,1)$ and$(1,1,1)$?
Please help.
3
votes
1answer
46 views
Eigenvalues and Eigenvectors Diagonilization
Let $ A=\begin{bmatrix}
-7 & -1 \\
12 & 0 \\ \end{bmatrix} $ . Find a matrix $ P $ and a diagonal matrix $D$ such that $PDP^{-1} = A$.
Ok so the first thing I need to look ...
0
votes
2answers
19 views
question on linear algebra-vector spaces-basis
Let $V$ be the space of $2X2$ matrices over $F$. Find a basis ${A_1,A_2,A_3,A_4}$} for V such that $A_j$$^2$=$A_j$ for each $j$?
Please solve this question i cannot solve it.
1
vote
1answer
36 views
Question on orthogonal subspaces
I'm given a problem with the initial values
$u =\begin{bmatrix}1\\1\\0\end{bmatrix} , v = \begin{bmatrix}2\\3\\0\end{bmatrix}, b =\begin{bmatrix}4\\5\\6\end{bmatrix}$.
I've calculated that the ...
1
vote
5answers
77 views
Linear Transformation from $ \mathbb R^2 \rightarrow \mathbb R^2 $
Let $ v_1 = \begin{bmatrix} 1 \\ -1 \\ \end{bmatrix} $ and $ v_2 = \begin{bmatrix} 2 \\ -3 \\ \end{bmatrix} $ Let $ \mathbb R^2 \rightarrow \mathbb R^2 $ be linear transformation satisfying $ ...
0
votes
1answer
67 views
Problem related to differential of a map
I dont understand how to solve this problem. Please can you explain the solution clearly? I want to learn how to solve such problems. Thank you
5
votes
4answers
126 views
Linear Transformations $ \mathbb R^2 \rightarrow \mathbb R^3 $
If $ T : \mathbb R^2 \rightarrow \mathbb R^3 $ is a linear transformation such that $ T \begin{bmatrix} 1 \\ 2 \\ \end{bmatrix} = \begin{bmatrix} 0 \\ 12 \\ -2 \end{bmatrix} $ and $ ...
0
votes
1answer
27 views
Basis of the orthogonal component
Let $\{v_1,\dots,v_n\}$ be an orthogonal basis for $R^n$ and let $W=\operatorname{span}\{v_1,\dots,v_k\}$. Is it necessarily true that $W^\perp=\operatorname{span}\{v_{k+1},\dots,v_n\}$? Either prove ...
0
votes
1answer
42 views
Column space of a $QR$-factorization
Let A be an $m$ x $n$ matrix with linearly independent columns and let $A=QR$ be a $QR$-factorization of $A$. Show that $A$ and $Q$ have the same column space.
I honestly don't have a clue where to ...
1
vote
2answers
36 views
linear algebra -vector spaces
If $V$ is a vector space over the field $F$ then verify that
$$(\alpha_1+ \alpha_2)+(\alpha_3+\alpha_4)=[\alpha_2+(\alpha_3+\alpha_1)]+\alpha_4$$
for all the vectors ...
1
vote
1answer
91 views
How does one go about proving that something is a vector space?
So I have this pretty theoretical problem for homework that says I need to show that this set of matrices is a vector space. It says that we have the set $M_{m,n}(\mathbb R)$, $m\times n$ matrices ...
0
votes
3answers
52 views
Basis of the subspace of $\mathbb R^4$
Find a basis of the subspace of R4 consisiting of all vectors of the form:
$$\begin{bmatrix}x_1\\
6 x_1 + x_2\\
4 x_1 + 5 x_2\\
8 x_1 - 9 x_2\end{bmatrix}$$
Now, I really have no clue how to set ...
0
votes
1answer
26 views
Height of tripod with two equal legs and one shorter leg
The legs of a tripod are at right angles to each other. Two legs have the same length $a$ and the third is longer with length $b$. The lower ends of the legs are placed on a level floor. What is the ...
1
vote
1answer
54 views
Find a set of vectors $\{u,v\}$ in $\Bbb{R}^4$ that spans the solution set
Question: Find a set of vectors $\{u,v\}$ in $\mathbb{R}^4$ that spans the solution set of the equations:
$$\begin{align}x - y - z + w = 0 \\
x + 2y - z + 3w = 0\end{align}$$
Reducing these I get:
...
0
votes
1answer
78 views
compute the similarity between two vectors
Euclidean distance is a measure that may be used to compute the similarity between two vectors. Given a query $q$ and documents $d_1, \ldots, d_n$, we may rank the documents $\mathcal{D} = ...
1
vote
0answers
47 views
Vector space-minimal polynomial
Let $V$ and $W$ be finite-dimensional vector spaces over $R$ and let $T_1 \colon V \rightarrow V$ and $T_2 \colon W \rightarrow W$ be linear transformations
whose minimal polynomials are given by ...
0
votes
1answer
173 views
Linearly Independent set of vectors that spans the same subspace of $\mathbb{R}^3$
I'm having trouble setting this up.
I have these $3$ column vectors:
$\langle 1, 1, 2\rangle$
$\langle -7, -1, -8\rangle$
$\langle 3, 0, 3\rangle$
I need to find a linearly independent set of ...
0
votes
4answers
217 views
Two linearly independent vectors perpendicular to vector $u$
I'm having trouble with these types of questions. I have the following vector $u = (4, 7, -9)$ and it wants me to find 2 vectors that are perpendicular to this one.
I know that $(4,7,-9)\cdot (x,y,z) ...
1
vote
1answer
45 views
Is there an error in “Mathematics for Physicists”, p. 107?
In deriving Bra notation and using it to define the scalar product in terms of the product of a Bra and a Ket, my text (p. 107) says that
...
2
votes
1answer
91 views
For what values of a is this vector in the span
I'm stuck on this one question that was on my math work sheet.
Say that a certain vector space $V$ consists of triples of real numbers $u=(u_1,u_2,u_3)\in \mathbb{R}^3$ with vector addition and scalar ...
1
vote
5answers
47 views
How can one complete a set to a vector basis?
What are the possible ways of solving next trivial task:
$$
\mathbf{u} =
\left( \begin{array}{c}
1 \\
2 \\
0 \\
\end{array} \right)
\mathbf{v} =
\left( \begin{array}{c}
5 \\
5 \\
2 \\
\end{array} ...
2
votes
1answer
40 views
On the existence of a bounded linear functional
Let $\mathcal{H}$ be a Hilbert space. By the Riesz Representation Theorem, we have that any bounded linear $\psi \in \mathcal{H}^{*}$ is of the form $\psi(h) = \langle h, g \rangle$ for some $g \in ...
1
vote
3answers
43 views
Differentiation operator and eigenvalues
Let $V = \{p(x) \in F[x] \ | \ \deg(p(x)) \le n\}$. Let $T : V \to V$ be given by differentiation, in essence $$T(p(x)) = p'(x)$$
It seems to me that the only eigenvalue that can exist is $\lambda ...
1
vote
1answer
145 views
Regarding the kernel of a linear transformation and that of the associated representing matrix
Let $V, W$ be finite dimensional vector spaces over a field $F$. Let $\mathcal{B}_{V} = \{\mathbf{v_1, \cdots, v_n} \}$ and $\mathcal{B}_{W} = \{\mathbf{w_1, \cdots, w_m} \}$ be corresponding bases. ...
