All Questions
17
questions
1
vote
1
answer
48
views
Proof that two matrices are row-equivalent iff they have the same nullspace
The matrices are both of size m x n over some field F, obviously.
The first direction of this proposition is clear enough, however the opposite direction (same nullspace -> row-equivalence) is ...
- 351
1
vote
0
answers
54
views
An exercise about vector multiplication/scalar product
I came across an exercise in vector multiplication but couldn’t find the formal answer for it to check myself.
I would really appreciate if someone could show me the correct answer for this exercise.
...
- 21
1
vote
1
answer
33
views
What is the explicit expression of a plane wave in the frequency domain?
A plane wave in the time domain can be written (using notation for an electric field):
$$\boldsymbol{E}(\boldsymbol{r},t)=\boldsymbol{E}_0 e^{i(\boldsymbol{k} \boldsymbol{r}-\omega t)}$$
what is the ...
- 187
1
vote
0
answers
77
views
Confusion with gradient and divergence of vector fields and scalar fields
In a proof, my teacher used the following assumption for an approximation:
$$
|\nabla n(\boldsymbol{r}, t)| \ll\left|\nabla \partial_{t} \boldsymbol{\tilde{r}}(\boldsymbol{r}, t)\right|
$$
where $n$ ...
- 853
0
votes
4
answers
209
views
Linear algebra: proving these vectors form a basis of $\mathbb{R}^{4}.$
Given $v_1 = (1, 0, 1, 2)^T$, $v_2 = (0, 1, 1, 0)^T$, $v_3 = (−1, 2,
1, 0)^T$, $v_4 = (0, 0, 1, 0)^T$.
Prove that $v_1, v_2, v_3, v_4$ form a basis of $\Bbb R^4$.
I have no idea because I thought ...
- 37
0
votes
0
answers
52
views
Rotation of an anisotropic inner product
Consider the following function $$\sum_{i=1}^3{ \frac{(q_i-d_i)^2}{a_i} },$$
which can be seen as an anisotropic inner product in the $\vec{q}$ vector space that is rescaled by $a_i$’s and offset by $\...
- 784
0
votes
2
answers
267
views
Find the normal vectors of the two angle-bisecting planes.
Compute the coordinate equation of the angle bisectors of the planes E
and F.
$E: x + 4y + 8z + 50 = 0 $ and $F: 3x + 4y + 12z + 82 = 0$
Proceed as follows:
a) Find the normal ...
- 51
1
vote
2
answers
482
views
Are there multiple types of "components" in the study of vectors?
In class we were taught how to write any vector in "component form" as $(a, b)$, where $a$ is "change in x" and $b$ is "change in y."
However, yesterday we were also taught how to "decompose" a ...
- 830
0
votes
2
answers
52
views
Vectors, calculate distance from these two points [closed]
My Attempt Thus far.. I tried using the quadratic formula on this to find the distance/magnitude , however it did not work. I then tried to solve it as a quadratic inequality.
Can anyone help?
And ...
0
votes
1
answer
59
views
How can I prove that $a=4$ in the algebraic equation $a+3=7$ using vectors and vector properties?
Consider the simple algebraic equation $a + 3 = 7$. The obvious answer is $a=4$. I was tasked with proving that $a = 4$ by using vector properties. It has something to do with the additive identity ...
- 119
0
votes
2
answers
105
views
True or false: For all $A \in \mathbb{R}^{n \times n}$ we get that det($A+A$) $= 2^{n}$ det($A$)
Let $n \in \mathbb{N}$
True or false: For all $A \in \mathbb{R}^{n \times n}$ we get that
det($A+A$) $= 2^{n}$ det($A$)
I did several tests and they all worked fine so I'd say that the ...
- 4,710
0
votes
2
answers
622
views
True or false? The linear system of equations $Ax=b$ is uniquely solvable for every $b \in \mathbb{R}^{n}$ if and only if det(A) $\neq$ 0 [duplicate]
Let $n \in \mathbb{N}$, let $A \in \mathbb{R}^{n \times n}$
True or false? The linear system of equations for $Ax=b$ is uniquely
solvable for every $b \in \mathbb{R}^{n}$ if and only if
$\...
- 4,710
0
votes
1
answer
1k
views
Three vectors are given, choose a basis for the subspace
The subspace $S \subseteq \mathbb{R}^{4}$ is spanned by the vectors
$v_{1}=\begin{pmatrix} 1\\ 2\\ 3\\ 0 \end{pmatrix},
v_{2}=\begin{pmatrix}
-1\\ 5\\ 7\\
-1 \end{pmatrix}, v_{3}=\begin{...
- 4,710
1
vote
1
answer
374
views
Figure Out Direction of Plane (Heading) — Slightly Complicated?
Bearing = Measured from Due North ($310^\circ$ bearing = $140^\circ$)
We are given the following — a plane departs from airport LAX to SFO at an air speed of 750 km/hr at 12:00 AM, with airport SFO ...
- 177
0
votes
1
answer
427
views
Find the value of |m-n|
The magnitude of two vectors $m$ and $n$ are 5 and 4 units respectively. The angle between $m$ and $n$ is 60 degrees.
How can I find $|m-n|$?
- 7,788