Tagged Questions
2
votes
2answers
51 views
A is a matrix of integers , prove that A+I is invertible
Question:
$2 \le d \in \Bbb Z$
Let $A \in M_n(\Bbb Q) s.t$ All of it's elements are integers divisible by d.
Prove that $I+A$ is invertible.
What I thought:
I thought of using the determinant of ...
1
vote
1answer
15 views
How to show that every complex matrix with orthonormal columns can be supplemented into an unitary matrix?
Show that every matrix $A \in M_{n,k}(\mathbb{C})$ whose columns are orthonormal vectors in $M_{n1}(\mathbb{C})$ can be supplemented with additional n-k columns to an unitary matrix $U \in ...
1
vote
1answer
17 views
Paralellogram based pyramid volume
Given three points of a paralellogram being
$
P(2,3,4) ; Q(3,1,4); R(2,5,3)
$
I've already calculated the fourth point with
$S(1,7,3)$
Further there's the tip point $Z(5,7,8)$
To get the volume of ...
0
votes
1answer
25 views
Extension of the field of scalars.
I want to make sure that I understand this correctly:
$V- \mathbb{Q}$ vector space.
Then $V_\mathbb{R} := V \otimes \mathbb{R} $
is naturally an R-vector space (next to
being a $\mathbb{Q}$-vector ...
8
votes
3answers
58 views
Prove that matrix is non-negative
Problem:
Given $A_{1}, A_{2}, ..., A_{n}$ - finite sets and $a_{ij} = |A_{i}\cap A_{j}|$ - number of elements in intersection of sets. Prove, that matrix $(a_{ij})_{i=1,2,..,n}^{j=1,2,.., n}$ is ...
0
votes
1answer
32 views
Cross product as factor in dot product [duplicate]
Given there are two vectors $w,v$ with $||w||=4$ , $||v||=1$ and $\phi=\frac{2\pi}{3}$
How do you transform the following expression into a form in which it can be computed with the given ...
1
vote
0answers
28 views
Basis of kernel and image of a linear transformation - verification
The transformation matrix I found is: $$\begin{pmatrix} 1 & -1 \\ 1 & 1 \\ 0 & 0\end{pmatrix}$$
Is this how a basis for $\ker$ and $\mathrm{im}$ is calculated?
$$\begin{pmatrix} 1 & ...
0
votes
2answers
42 views
Find the relation between the dimension of the nullspace of $A$ and $A^t$
Let $A$ be a $n \times n$ matrix, what is the relation between the dimension of the nullspace of the homogeneous system of $A$ and the one of $A^t$?
0
votes
1answer
29 views
$U\subseteq V$ is $T$ invartiant $\Rightarrow$ $\left(T\,|_{U}\right)^{*}=\left(P \circ T^{*}\right)\bigl|_{U}$
I want to proove that given $T\in\mathcal{L}\left(V,V\right)$ ($V$ is a finite dimensional inner product space) and a subspace $U\subseteq V$ which is $T$ invariant that
...
0
votes
2answers
58 views
Computing cross product using norm and angle
Sorry for the weird title, if someone finds a better title for my problem be my guest to edit it ;)
For $\mathbf{v,w} $ in R³ with $\mathbf{||v||=1 ;||w||=4; \theta
=\frac{2\pi}{3}}$
Solve ...
2
votes
1answer
39 views
If $V$ is a vector space over a division ring $K$, and $A=\mathrm{End}_K(V)$, then every quotient ring of $A$ is a prime ring
Let $K$ be a division ring, let $V=V_{K}$ a vector space over $K$, and let $A=\mathrm{End}_{K}(V)$. Could anyone give me an idea of how to prove that every quotient ring of $A$ is a prime ring?
1
vote
1answer
32 views
Odd and even functions- a direct sum?
Question:
Let V be the vector space of all functions $\Bbb R\to \Bbb R$.
Show that $V=U \oplus W$
for $U=${$f | f(x)=f(-x) \forall x$}$, $W={$f | f(x)=-f(-x) \forall x$}
What I did:
I did prove ...
0
votes
1answer
41 views
Homework involving unitary diagonalization
I was given as an assignment to diagonalize the following matrix:
$\left(\begin{array}{cc}
\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta
\end{array}\right)$
I started by finding ...
0
votes
0answers
18 views
Distance between two affine lines using determinant of Gramian matrix.
I've a task to find the distance in $E^4$ between:
$L = [1,2,-1,4] + \text{lin}((1,2,-1,0))$
and
$M = [2,3,1,5] + \text{lin}((2,1,0,2))$
My efforts to find the correct solution:
Let
...
6
votes
2answers
124 views
does a matrix like this exist?
Question:
Does a matrix $A \in M_{3 \times 3}(F)$ exist s.t. $A^4=
\begin{bmatrix} 0&0&1\\0&0&0\\0&0&0\end{bmatrix}$
What I thought:
I think it doesn't. How do you start a ...
3
votes
1answer
40 views
About $\mathcal{L}(V,W)$
Let $V,W$ are two vector space and let $S\subseteq V$. Define: $$S^{0}=\{T\in\mathcal{L}(V,W)\mid~T(x)=0, \forall x\in S\}$$ The problem aks me to verify $S^{0}$ is a subspace of $V$ and if $V_1,V_2$ ...
1
vote
0answers
28 views
Using a matrix to organise values into groups
Let's say I have a matrix of size 6 x 6.
Six students are 'ranking' six other students (including themselves). If I wanted to organise them into let's say, groups of three without picking and ...
3
votes
2answers
30 views
Matrix multiplication related to complex numbers?
Evaluate and simplify the product
$\begin{bmatrix} r\cos(\alpha) & -r\sin(\alpha) \\ r\sin(\alpha) & r\cos(\alpha)\\ \end{bmatrix}$ $\begin{bmatrix} s\cos(\beta) & -s\sin(\beta) \\ ...
4
votes
1answer
31 views
How to show that a valid inner product on V is defined with the formula $[x, y] = \langle Ax, Ay\rangle $?
Let $A \in L(V,W)$ be an injection and $W$ an inner product space with the inner product $\langle \cdot,\cdot\rangle $. Prove that a valid inner product on $V$ is defined with the formula $[x, y] = ...
3
votes
1answer
36 views
Prove if we have a square unitary Matrix $Q$, then $\det(Q) = e^{i\theta}$
Prove if we have a square unitary Matrix $Q$, then $\det(Q) = e^{i\theta}$
Using $\det(Q)\det(\bar{Q}^T) = I$, I get to the stage $\det(\bar{Q})\det(Q)=1$, but can't do much else with it.
Thanks for ...
1
vote
2answers
27 views
Matrix involving values of polynomials
I've been doing this problem but im stuck.
Be $f_1 f_2 f_3 \in \mathbb{R}_2$[$x $]. Proove that {$f_1$,$ f_2$,$ f_3$} form a base of $\mathbb{R}_2$[$x $] as $\mathbb{R}$ vector space, if and only ...
3
votes
0answers
49 views
Eigenbasis of a Hilbert space: isomorphism
Let $K$ be a matrix containing the dot product between points in a Hilbert space $\mathcal{H}$ (assume that it is finite-dimensional). Then, we could form a basis using the eigenvectors of a normal ...
1
vote
1answer
45 views
Dual basis and annihilator problem
I think they're fairly simple but I really don't know where to start, the problems are these:
First one:
$V$ a vector space of dimension $n$ and $\phi \in V^* \setminus \{0\}$.
Prove that $\dim ...
1
vote
3answers
28 views
notation question (bilinear form)
So I have to proof the following:
for a given Isomorphism $\phi : V\rightarrow V^*$ where $V^*$ is the dual space of $V$ show that $s_{\phi}(v,w)=\phi(v)(w)$ defines a non degenerate bilinear form.
...
0
votes
1answer
26 views
On finding adjoint of transformation.
Let $V$ be an inner product space and $v,w\in V$ be fixed vectors. Define $T(u)=(u,v)w$. How to find the adjoint mapping $T^*$?
1
vote
1answer
30 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
43 views
Prove or disprove that Y = AX-C
Let $A$ be an $m \times n$ matrix such that $\mathrm{rank}(A) = n \le m$. Prove, or disprove using a counter example:
Every $m\times n$ matrix $Y$ has a decomposition $Y = AX-C$, where $X$ and ...
2
votes
2answers
114 views
How to show that there exists a scalar $\lambda$ so that $A = \lambda I$? [duplicate]
Let $V$ be a finite dimensional vector space, $V \ne \{ 0 \}$ and $A\in L(V)$ an operator which commutes with every operator from $L(V)$. Show that there is a scalar $\lambda$ so that the following ...
1
vote
2answers
91 views
How to show that if $p(A) = 0 \implies p(\lambda_0)=0$?
Let $V$ be a finite dimensional vector space and $V \ne \{ 0 \}, A\in L(V), \lambda_0 \in \sigma(A)$. If $p(\lambda)$ is an arbitrary polynomial for which the following applies: $p(A) = 0 $, prove ...
0
votes
0answers
27 views
Kernel and image of a set of linear maps [duplicate]
I'm having some trouble with this question (I think I didn't reach an acceptable answer - also hope I used the correct terminology).
Let $V,W$ be sets of vectors over $F$ of finite dimensions. We'll ...
0
votes
1answer
25 views
Projection transormation, proof of existence and uniqueness
Question:
A Projection Transformation is defined to be a linear transformation from V to V that satisfies $T^2=T$.
Let $V=U \oplus W$ . Show that there exists only one linear projection $T:V \to V$ ...
2
votes
1answer
52 views
Finding a kernel of a linear transformation of linear transformations.
Question:
Let V,W be vector spaces over field F.
We mark L(V,W) as the vector space of linear transformations from V to W.
Let $v_0 \ne 0$. We define a transformation: $\Psi: L(V,W) \to W$ that sends ...
0
votes
3answers
47 views
If the union of $A$ and $B$ is linearly independent then the intersection of the spans $= \{0\}$
$\newcommand{\sp}{\operatorname{sp}}$ Let $V$ be a vector space over $F$ field, and let $A,B$ be two different, disjoint, non-empty sets of vectors from $V$.
Prove or disprove the following:
...
1
vote
3answers
45 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 ...
2
votes
3answers
104 views
If $\operatorname{sp}(A) \cup \operatorname{sp}(B)=\operatorname{sp}(A\cup B) \Rightarrow A\cup B$ is linearly dependent
$\newcommand{\sp}{\operatorname{sp}}$
Let $V$ be a vector space over $F$ field, and let $A,B$ be two different, disjoint, non empty sets of vectors from $V$.
If $\sp(A) \cup \sp(B)=\sp(A\cup B) ...
0
votes
1answer
22 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
0answers
34 views
I'm having trouble finding this matrix $T$ relative to $\mathcal B$ and the standard basis $\mathcal E$ for $\mathbb R^2$
This was a homework assignment, but unfortunately it was the last homework assignment of the semester so I never got feedback and I'm just reviewing it for a final. I'm supposed to let $\mathcal ...
4
votes
2answers
58 views
Positive semidefiniteness of a block matrix of positive semidefinite matrices
Given any symmetric matrix $\mathbf{M} = \begin{pmatrix}
\mathbf{A} & \mathbf{B}\\
\mathbf{B}^\mathrm{T}& \mathbf{C}
\end{pmatrix}$, the following conditions are equivalent:
(1) ...
1
vote
2answers
23 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 ...
5
votes
3answers
312 views
REVISITED$^2$: Solution in $\mathbb{R}^n \overset{?}{\implies}$ Solution in $\mathbb{Q}^n$
Let $A ∈ M_{m\times n}(\mathbb{Q})$ and $B ∈ \mathbb{Q}^m$. Suppose that the system of linear
equations $AX = B$ has a solution in $\mathbb{R}^n$. Does it necessarily have a solution
in ...
1
vote
0answers
35 views
Determinant of a matrix with variables in it
Assuming that $z \neq 0$, compute the determinant $d_n(z) = \det D_n \left(1, z, 1 - \frac{1}{z^2} \right)$, where $z$ is a complex variable. In particular, compute the value $d_n(\sqrt{2})$.
...
0
votes
2answers
69 views
Topological manifold example
$\theta(x,x^2)=x$
$\Bbb X =${$(x,x^2)| x$ in $\Bbb R$}
And V is subset of $\Bbb R$
$dim\Bbb X=1$
My instructor said that this is topological manifold.
Why?
Please can you explain me? This ...
1
vote
0answers
41 views
$f: E^3 \rightarrow E^3$ is an isometry, and $\det f = 1$ and $f'\neq id$
Suppose, that $f: E^3 \rightarrow E^3$ is an isometry, and $\det f = 1$ and $f'\neq id$
Please help me prove, that $f$ is a composition of rotation about an axis and moving along this axis.
I don't ...
0
votes
2answers
19 views
Solving the simultaneous equations
I need to solve the following simultaneous equations for the constants $A$ and $B$:
$$zA + (1-z)B = 1$$
$$\hspace{1.8cm} z^2A + (1-z)B = z^2 - z + 1$$
where$z$ is just a variable. What I did was ...
5
votes
0answers
42 views
Prove that if $Q^tQ = I$ and $A = QR$, then $\|Ax - b\| = \|Rx - Q^tb\|$
I have a linear algebra final tomorrow and was practicing a few proofs. I want to make sure this proof is correct.
Prove that: If $Q^tQ = I$ and $A = QR$, then $\|Ax - b\| = \|Rx - Q^tb\|$
...
8
votes
1answer
54 views
If $A$ is orthogonal, for any vector $x$ such that $Ax = b$, $\Vert x \Vert = \Vert b \Vert$
Is this statment true: For any vector $x$ such that $Ax = b$, $\Vert x \Vert = \Vert b \Vert$, if $A$ is orthogonal.
I was working on a proof for my linear algebra class, when I noticed that the ...
2
votes
1answer
21 views
finding dual basis of vector space of polynomial degree less than or equal to 3
Assume that $V = P(3)$, Describe a basis for $V^*$ and express the linear functional $f : V \to \Bbb R$ given by $f\left(a_3x^3 + a_2x^2 + a_1x + a_0\right) = 2a_3 + a_2 - 5a_0$ as a linear ...
1
vote
2answers
61 views
Proving that a group is closed under multiplication by scalars
I need to prove that the group $U = \{A \in M_n(\mathbb{R})\mid AB=BA\}$ is closed under multiplication by scalars.
So I let $\alpha \in \mathbb{R}$ and let $A \in U$, which means I need to prove ...
3
votes
3answers
101 views
Show that the matrix $A^2 + I$ is invertible for all matrices $A$, where $A$ is an $n \times n$ symmetric matrix.
I'm a little stuck on this problem. I know that since $A$ is symmetric, $A=A^{T}$. I'm also pretty sure that $AA^{T}$ is invertible. Therefore $A^2$ would be invertible. I'm not really sure how to ...
0
votes
3answers
48 views
Problem with forces
A motor boat with the power to steer across a river at $30$ kmph is moving such that the bow is pointed in a northerly direction. The stream is moving eastward at $6$ kmph. The river is $1$ km ...
