0
votes
2answers
35 views

Transformation Matrix project

My task is to find the Transformation Matrix, that projects, any point of the xy-plane, on the line $$ y = 4x$$ The solution should be: $$T=\pmatrix{0.06&0.235\\0.235&0.94}$$ But somehow i ...
1
vote
1answer
29 views

How to find an unitary transformation of $A$ that minimize $(A'_{i,i}-1)^2$?

Is there a way to find an unitary transformation $$ A'=U^+AU $$ that minimize: $$(A'_{i,i}-1)^2$$ In other words, the diagonal elements must be similar to one: $A'_{i,i} \approx 1$ Any hint? ...
0
votes
1answer
30 views

Linear transformation

Let there be a linear transformation $T:R^3\rightarrow R^2$ Is there a linear transformation so that: $Ker(T)=Span((1,2,1),(0,3,-1))$ and $Im(T)=Span((5,-7))$ Answer: $Dim(V)=Rank(T)+Null(T)=2+1=3$ ...
0
votes
2answers
19 views

Writing a composite transformation as a matrix multiplication

I am confused about a question on matrix multiplication of a transformation. I have two matrices, P and Q as follows: $$P = \begin{pmatrix}\frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} ...
0
votes
0answers
22 views

What is the proof/show that the post of linear transformation generated by LDA is at most k-1

What is the proof/show that the matrix $Sw$ generated by LDA is at most rank $p-k$, where $p$ is the dimension of the data and $k$ is the number of classes. LDA: ...
0
votes
1answer
21 views

Transformation of a surface normal

I'm taking a university level course in discrete geometrics and graphical programming, and I'm having trouble understanding this exercise. Let p be a point in R^3, n a surface normal, and M a ...
0
votes
0answers
15 views

help me find the gimbal locks

I have this transformation (x, y, z) |-> (x'', y'', z''). How can the gimbal locks be discerned and where are they? ...
1
vote
0answers
72 views

Trick for Jordan-Matrix and transformation of basis

some time ago I found a 'trick' for getting a basis-transformation-matrix for jordan. I'd like to understand it, but at a certain point I stuck. Maybe you can help me? Given is a matrix A: ...
0
votes
0answers
12 views

Extract translation vector from two homogenous transformation matrices

Given two homogenous transformation matrices $$ A = \begin{pmatrix} a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ ...
0
votes
0answers
42 views

How do you solve a linear transformation with no transformation matrix given?

I am stuck, I can't see how Tff was found with no transformation matrix. And now am being asked to find Tgg, help me http://oi60.tinypic.com/33yrplv.jpg
0
votes
1answer
43 views

I need help with linear transforms? Linear Algebra [closed]

In the question below, how was [T]ff found? I have tried but I can't understand how because I usually start from a given matrix with variables, but non is given here. website is here; ...
0
votes
2answers
20 views

Transformation matrix of a polynomial

I would really appretiate some help about the following transformation matrices. We have to write a tranformation matrix in basis $B = \{ 1 + x, x + x^2, x^2 \}$ with a polynomial $(Ap)(x) = (x^2 - ...
0
votes
1answer
19 views

Transform gradient to reference element

Minimal example of the problem How can you transform the gradient to the reference element?
1
vote
1answer
83 views

Given $A$, find invertible $B$ such that $B^{-1}AB$ is positive

Given $A \in Mat(n,n,\mathbb R)$, is there always an invertible matrix B, such that $B^{-1}AB$ is positive, assuming all eigenvalues of A are positive and simple ? If yes, is it possible to classify ...
0
votes
1answer
24 views

Getting linear combinations in linear algebra?

I failed a homework problem a few days ago. I can't figure out how they got the answers, which have been given in green as corrections. Help me figure how they got them;
1
vote
1answer
41 views

Constructing regular integer matrices with distinct integer eigenvalues

How can I construct matrices with positive integer values and distinct integer eigenvalues (not necessarily positive, but 0 should not be an eigenvalue). The standard-method to construct matrices ...
3
votes
2answers
76 views

Show $rk(A) + rk(B) \ge rk(A+B)$

Show $rk(A) + rk(B) \ge rk(A+B)$, where $A,B \in M_{m\times n}(\mathbb{F})$ I'm trying to think in terms of linear transformations. We can define $T_a, T_b:\mathbb{R}^n\rightarrow \mathbb{R}^m$ I ...
1
vote
1answer
35 views

Matrices as linear transformations

I am reading a proof which claims: A matrix of $m\times n$ is a linear transformation from $m$ vector-space to $n$ vector-space, And therefore, by the dimension theorem: $m = \dim\ker A + ...
0
votes
1answer
34 views

Matrix transformations

I have to find components of a matrix for 3D transformation. I have a first system in which transformations are made by multiplying: $M_1 = [Translation] \times [Rotation] \times [Scale]$ I want to ...
1
vote
1answer
46 views

Linear transformation matrix representation with differentiation answer confirmation

I hope you liked the title. I have a question that is as follows: Consider the linear transformation $T: P_3(\mathbb{R}) \to P_3(\mathbb{R})$ given by $$T(f(x))=f(0)+f'(x)+f''(x)$$ Where the ...
4
votes
1answer
89 views

Finding a basis for $\ker(T)$

I have this question: Let $Z\in M_{2\times2}(\mathbb{R})$ be defined as $$Z = \left( \begin{align} 1 &&1\\1 &&1 \end{align} \right)$$ and consider $T: ...
0
votes
1answer
25 views

Lorentz transformation and Minkowski metric

For the exam I'm trying to solve some problems. Today I found this exercise and need some help: For the group S0(1,1) of the Lorentz transformation I have $\phi \in \mathbb{R}$ and $A_{\phi}: ...
0
votes
1answer
49 views

Do T and T* have the same eigenvalues with the same algebraic multiplicity?

I know that the eigenvalues of T* are the conjugates of T's eigenvalues , but how can I see each eigenvalue of T and it's conjugate , the eigenvalue of T*, have the same algebraic multiplicity?
1
vote
2answers
30 views

trace function ($2\times2$) with ordered bases as linear transformation

We got trace function as following: $$\operatorname{tr}\begin{pmatrix} a & b\\ c & d\\ \end{pmatrix}=a+d$$ So now have to write down $[\operatorname{tr}]_{S_1,S_2}$, ...
0
votes
0answers
19 views

Find the reference point required to transform scale two elements uniformly

This is actually a programming issue I am having but the answer is rooted in matrix mathematics so this seems like the best place to ask it. I am no mathematician so I apologise if some of my concepts ...
0
votes
1answer
20 views

dimension of kernel and image of isomorphism

T:V ->V is isomorphism, dim V = n. The kernel of isomorphism has only vector 0 in it, so by rank nullity theorem does it mean that dim of kernel is 1 and dim of image is n-1? the question seems a ...
0
votes
0answers
25 views

Transformation matrices and hermitian/unitary/normal/… matrices

I need some help with the following - have I done the correct things or how can I solve the task? Let $f \in End(V)$, V a unitary space $\mathbb{C}^3$ given by: $A_{\alpha \beta} (f) = \frac{1}{7} ...
0
votes
1answer
19 views

Show there's no ordered basis $E$ with the following conditions

Let $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$ such that: $$T\left( {\matrix{ x \cr y \cr } } \right) = \left( {\matrix{ 2 & 1 \cr 3 & 4 \cr } } \right)\left( {\matrix{ ...
1
vote
1answer
26 views

Equation of matrices

Let $V$, a 3d vector space above $F$. Let $T:V\rightarrow V$, linear transformation and $E$, an "ordered" basis such that: $$[ T ]_E = \left( \matrix{ 0 & 0 & a \cr 1 & 0 & ...
2
votes
0answers
15 views

Finding the matrix ${\left[ T \right]_E}$

Let the matrix ${\left[ T \right]_{B \to E}}$, the matrix where: $${\left[ T \right]_{B \to E}}{\left[ v \right]_E} = {\left[ {T(v)} \right]_B}$$ It's given that: $${\left[ T \right]_{B \to E}} = ...
2
votes
0answers
72 views

Finding transformation from $T : \Bbb R^5 \rightarrow \Bbb R^4 $ …

Is there a Linear Transformation from $T : \Bbb R^5 \rightarrow \Bbb R^4 $ so $$\operatorname{Ker}T = \{( x,y,z,t,w) \in \Bbb R^5 \; | \; x = 2y, \text{ and, } z = 2t = 3w\}$$ if so find an example of ...
1
vote
1answer
66 views

Calculating the adjustment translation to be applied after rotating and scaling so that operations pivot about a given point.

I have a matrix for transforming an image into a target frame. The matrix is a function of a scale, $s$ rotation angle, $\theta$, and a translation that is applied after rotating, $tx, ty$. The ...
1
vote
0answers
21 views

Linear Transformation - linear algebra question [duplicate]

$T:\mathbb{R}_2[x] \mapsto \mathbb{R}_2[x]$ s.t.: $$ \begin{array}{l} T(1) = 3+2x+4x^2, \\ T(x) = 2+2x^2, \\ T(x^2) = 4+2x+3x^2. \end{array} $$ Is there base $B$ of $\mathbb{R}_2[x]$ that $[T]_B = ...
1
vote
1answer
46 views

linear transformations with matrices $A, A^*$

Let $K$ be a field, $K\subseteq \Bbb C$. $V$ is a linear space over $K$, $\dim(V)=n(n\geq2)$. Choose ordered basis $\epsilon_1,\epsilon_2,\dotsc,\epsilon_n$ for $V$. $\bf A,B$ are two linear ...
1
vote
0answers
40 views

Matrix for orthogonal projection with respect to ordered and canonical bases

Orthogonal projection onto the line $y = 2x$ gives a linear transformation $T: R2 → R2$ such that $$T(1,2) = (1,2)$$ and $$T(−2,1) = (0,0)$$ Then the matrix of T with respect to the ordered basis ...
1
vote
1answer
40 views

matrix representation of linear transformation

For a set $N$ let $id_N:N \rightarrow N$ be the identical transformation. Be $V:=\mathbb{R}[t]_{\le d}$. Determine the matrix representation $A:=M_B^A(id_V)$ of $id_V$ regarding to the basis ...
0
votes
0answers
51 views

Help determining whether a function is a linear transformation $T:M_{2,2}\rightarrow R, T(A)=|A|$

Again, here is the function: $T:M_{2,2}\rightarrow R, T(A)=|A|$ I was able to prove that its not a linear transformation because $T(A+B) \neq T(A)+T(B)$ in fact, $T(A+B) = C$ where $C$ is a new ...
2
votes
1answer
68 views

Meaning of entries of a transformation matrix in practical terms [Homework related]

I'm having a bit of trouble understanding what the matrix entries mean practically in this problem: 100 kg of a highly toxic substance is spilled into three lakes. The state, t weeks after the ...
0
votes
1answer
19 views

Matrix representations of linear transformation between bases

Let V and W be vector spaces, and let L: V -> W be a linear transformation between them. A basis for V is E = {$v_1$,...,$v_5$}. A basis for W is F = {$w_1$,...,$w_4$}. On the basis vectors the linear ...
0
votes
1answer
89 views

Looking for a formula to calculate DCT/FFT frequencies when cropping a matrix/image.

Given: A is a matrix of dimensions W1 x H1 . Cropping: Few rows and/or few columns were deleted from matrix A. We got matrix B of dimensions W2 x H2. Not more than 5% of matrix A rows/columns ...
0
votes
0answers
34 views

How to calculate the combined frequencies of a DCT matrix?

Given a 2D matrix of dimensions w1,h1. I preform a DCT 2D transform on the matrix (DCT = DCT type 2). I get a 2D result matrix. This matrix has two frequency axes - x,y (which are simply the ...
0
votes
0answers
45 views

Using basic transformations to derive matrix for the reflection in a line?

Using basic transformations (translation, scaling and rotations), show all the steps to derive the transformation matrix for the reflection of points n the line : y = 3 - x I know that a directional ...
2
votes
1answer
95 views

How does a cropping of a 2D matrix/image affect its DCT transform?

I apologize in advance: since I am not a mathematician, maybe my question is not well defined, but I hope that some of you will still understand my meaning. Given a 2D matrix, or an image of ...
0
votes
1answer
21 views

Rotation operator for a point in a coordinate system linearly derived from Cartesian coordinates

For some experimental and practical reason, I have created a new coordinate system in the form $$x^\prime_i=T_{ij}x_j$$ where $T_{ij}$ isn't a square matrix. $x_i$ is standard Cartesian coordinates, ...
1
vote
0answers
21 views

Rigid Deformation

I'm trying to parse through this paper on using the method of moving least squares for rigid transformations - http://www.cs.rice.edu/~jwarren/research/mls.pdf Under section 2.3, the author mentions ...
0
votes
1answer
102 views

How to find the rotation matrix that will align an arbitrary vector to an axis

If I have a vector that starts at the origin, how can I find the transformation matrix that will align it with the positive y-axis. So it basically turns into a positive-y axis? EDIT: I also forgot ...
0
votes
1answer
30 views

Matrix Transformation - Using matrix multiplication

How do I use matrix multiplication to find the reflection of (-1,2) about the x axis, y axis and the line y=x?
0
votes
0answers
13 views

Inverse coordinates on a matrix

I am trying to "inverse" some coordinates on a matrix. For example, take this grid: ...
1
vote
2answers
137 views

Transformation matrix from quadrilateral to rectangle

There exists a rectangle somewhere in space with some orientation. A camera from the coordinate center point is looking along the z axis and is seeing the rectangle as a quadrilateral (due to ...
1
vote
1answer
41 views

Transformation Matrix $M_B^B$ of $P_3$ for $B = (1,x,x^2,x^3)$. Is that correct?

I have the following task and just wanted to check weather this is (written) correct(ly). Let $V$ be the vector space of all polynomials of grade $\le 3$ and $f: V \rightarrow V, p \rightarrow p'$ an ...