0
votes
0answers
9 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
16 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
0answers
12 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
21 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
14 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
11 views

Inverse coordinates on a matrix

I am trying to "inverse" some coordinates on a matrix. For example, take this grid: ...
1
vote
0answers
21 views
+50

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
29 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 ...
0
votes
1answer
23 views

Image of $\phi: \mathbb{Q}^{2\times 2} \rightarrow \mathbb{Q}^{2\times 2}, \ A \rightarrow A + A^t$

This question is related to the question I previously asked: Kernel. The following function is given: $$\phi: \mathbb{Q}^{2\times 2} \rightarrow \mathbb{Q}^{2\times 2}, \ A \rightarrow A + A^t$$ ...
1
vote
3answers
36 views

Help on finding eigenvalues of transformation on matrices

T is linear transformation working on 2x2 matrices: T(A) = $\begin{bmatrix}1 & 1\\1 &1\end{bmatrix}$ A as far as I see only 0 is an eigen value but someone told me 2 is eigen value too and ...
1
vote
2answers
56 views

Nilpotent Mappings

Got completely confused with this nilpotent and JCF stuff, need some help. Matrix $A_{n\times n}$ is nilpotent of order K, $1\le k\le 4$ Need to find: a list of all possible dimensions of ...
2
votes
2answers
62 views

Isomorphism between symmetric and upper triangular matrices

Question: Determine if the vector spaces $V=S_{3}$, the 3x3 symmetric matrices, and $W=U_{3}$, the 3x3 upper triangular matrices, are isomorphic. If they are, give an explicit isomorphism $T: V ...
1
vote
1answer
32 views

Linear Transformation from $\alpha$ to $\beta$

T: $R^3$ $\to$ $R^2$ $$[T]_{\beta\alpha} = \begin{matrix} 2 & 3 & 1 \\ 1 & 2 & 1 \\ \end{matrix} $$ $\alpha$ = {(1, -1, 1), (0, 1, 0), (1, 0, 0)} ...
0
votes
1answer
24 views

$3D$ projection onto a plane

I have an engineering problem involving math so I figured I ask it here. I have two sets of data: Acceleration in $3$ dimension is given by $\langle X,Y,Z \rangle $. Change of orientation along ...
0
votes
3answers
25 views

Composite linear map Rank and Image

I have been pondering on this question, I did part $(a)$ wherein you had to prove that $\operatorname{Im}(T)= \operatorname{Im}(T^{2})$ , but I am struggling to get the concept of part $(b)$, any help ...
2
votes
1answer
32 views

How can I combine affine transformations into one matrix?

So from what I understand from this picture, the box is stretched to twice its width. And it is then flipped from the x-axis. And then it is rotated 30 degrees anticlockwise. So these three ...
0
votes
0answers
14 views

Find angle-preserving transformation matrix given 2 points

I asked a similar question yesterday about finding an affine transform matrix given the same 2 points in both coordinate systems. I was told that there was only a unique solution, if the scaling was ...
0
votes
1answer
35 views

Find 2D affine transform matrix given a pair of points

I have the coordinates of two points in an initial 2d coordinate system and the corresponding coordinates in a target system. Is is possible to determine the affine transform matrix from these values? ...
1
vote
1answer
17 views

Function that transforms a Matrix to different dimensions

What is the name of a function that transforms a matrix into different dimensions? Say I have a matrix M of dimensions $(x,y)$ and I want to transform it to dimensions $(w,v)$. I can accomplish this ...
0
votes
1answer
23 views

Constructing a similarity matrix between points

I have two images with two sets of corresponding points. In order to align the images I'm trying to compute the similarity matrix that describes the relationship between the corresponding points. I ...
0
votes
0answers
16 views

Rotate the point P(2,1,1) 90 degree about the axis (Px=1, Py=1) parallel with the z-axis?

a) Rotate the point P(2,1,1) 90 degree about the axis (Px=1, Py=1) parallel with the z-axis? For this question I really don't know how to parallel the matrix. b) Reflect the point P (-1,2,2) ...
0
votes
0answers
26 views

3D transformation (matrix)

(a) Write down the one point perspective transformation (homoogeneous) that corresponds to an observer at the point y=3 looking down the y-axis? I don't know what this question is asking by ...
0
votes
0answers
35 views

Determinant of a transformation matrix

I have been reading about determinants and transformation matrices. After that I was reviewing some exercises on a book I got. In one exercise I'm asked to find the transformation matrix and the ...
1
vote
3answers
34 views

Linear Algebra Vector Space matrix help

Let $M_{2\times2}$ be a vector space of all $2\times2$ matrices. If the transformation from $M_{2\times2}$ to $M_{2\times2}$ is $t(A)=A+A^T$ and $A$ is a $2\times2$ matrix with the top row $a,b$ and ...
0
votes
1answer
28 views

Describing Cartesian transformations in Cylindrical Polar Coordinates

I have a question about converting functions defined in Cartesian coordinates to a cylindrical polar system. The particular coordinate transformation that I'm reading about is: \begin{equation} ...
1
vote
0answers
38 views

Linear Algebra Matrix Transformation Question

Can someone please help me out with this question. If a nonzero matrix $A$ is transformed from $\mathbb{R}^3$ to $\mathbb{R}^2$, then the null space of $A$ must be a one dimensional (sub)space of ...
1
vote
0answers
57 views

Algorithm to determine matrix equivalence

I'm a physicist who's not particularly good at linear algebra so please accept my apologies if this is standard textbook stuff that I'm just unaware of. I have two real rectangular matrices $A_{mxn} ...
1
vote
0answers
35 views

Describing transformations using base vectors

So I just learned that we can describe vector transformations of shapes using base vectors, where the base vector I = $$ \begin{pmatrix} 1 \\ 0 \\ \end{pmatrix} $$ and J=$$ ...
4
votes
2answers
143 views

Prove that the following matrices cannot represent the linear transformation $T$ in ANY basis

$T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ defined as $T(x,y,z) = (2x,z,y)$ is a linear transformation. I need to prove that the following matrices cannot represent $T$ in ANY basis: ...
0
votes
0answers
8 views

composition of multiple scales

i want to scale a point multiple times. Let P(x, y) be the point that is to be scaled, S the scale matrix and T the translation matrix. To scale the point P according to a point Q(x,y) you have to do ...
1
vote
1answer
70 views

Combine a rotation matrix with transformation matrix in 3D (column-major style)

I'm trying to concatenate a rotation matrix with a 4x4 homogeneous transformation matrix with column-major convention. I want this rotation matrix to perform a rotation about the X axis (or YZ plane) ...
0
votes
2answers
164 views

Consider the trace map $M_n (\mathbb{R}) \to \mathbb{R}$. What is its kernel?

The map is the trace map. I.e, it takes any $n$ by $n$ matrix and associates to that matrix, a number of the form $\mathrm{Tr}(A) = \sum_{i=1}^n a_{ii}$, where $A \in M_n (\mathbb{R})$. I need to ...
2
votes
2answers
43 views

$T: \mathbb R^n \to \mathbb R^n$, $\langle Tu,v\rangle=\langle u,T^*v\rangle$, is $T^*=T^t$ regardless of inner product?

Basic question in linear algebra here. $T$ is a linear transform from $\mathbb R^n$ to $\mathbb R^n$ defined by $T(v)=Av$, $A\in \mathrm{Mat}_n(\mathbb R)$. We are given some inner product $\langle ...
0
votes
0answers
85 views

$3$D transformation matrix to $2$D matrix

I have a $3$D affine transformation $4\times 4$ matrix. I need to convert it (project) to a $2$D affine transformation $3\times 3$ matrix, which looks like this: $3$D rotations are irrelevant and ...
1
vote
1answer
29 views

For general non-symmetric square matrices is there a matrix norm that is invariant under similarity transformations?

I think that there is no similarity-invariant matrix norm for general matrices. But are there similarity invariant norms for special types of matrices (e.g. for matrices whose eigevalues are different ...
0
votes
1answer
30 views

finding this linear transformation

i am following this guide: http://www.calpoly.edu/~brichert/teaching/oldclass/f2002217/handouts/goof.pdf my question is to find the linaer transformation that adheres to $T(1,1,1) = (1,1,1)$ ...
0
votes
1answer
45 views

Find rotation angle of given image

At first: our aim is to find the total transformation of left house to the right house. What I did it first is translating the house with the center to the origin. I already found out that the ...
0
votes
0answers
52 views

How does orthonormal basis rotating work?

When you insert an orthonormal set into the column vectors of a matrix, you create a rotation matrix. I can't understand how this works, by simply placing the the vectors in there you have a rotation ...
5
votes
1answer
63 views

How to compute (and check) this transform matrix?

Background: This is a homework exercise which asks to compute a transform matrix. The answer has been published by our teacher. However, my approach goes a different way and gets a different solution. ...
0
votes
5answers
123 views

Is it true that every orthogonal transformation , even over $\mathbb R$, is diagonalizable?

Is it true that every orthogonal transformation , even over $\mathbb R$, is diagonalizable? I didn't succeed to get any information about it. Could anyone explain please?
0
votes
1answer
52 views

Jacobian of an inverse

Suppose that we have an invertible map $T(u,v)=(x,y)$. The Jacobian of $T$ is given by $ \text{Jac}(T)= \left| {\begin{array}{cc} x_u & x_v \\ y_u & y_v \\ \end{array} } ...
0
votes
1answer
150 views

How do I find transformation matrix with respect to standard basis?

I know that in order to find transformation matrix with respect to a basis, I need to apply the transformation to said basis and the result is the column of the transformation matrix. But what ...
3
votes
1answer
64 views

Show that T is a linear transformation and find a, b, c

I'm having trouble understanding this question and the proper way to solve it. I don't understand the solution given and why this was the right way to answer it. Problem: For the vector space ...
0
votes
0answers
33 views

$K$-linear transformation $f:Mat (3 \times 2, K) \rightarrow K^3$

Could somebody please check my solution? Let $K$ be a field. Given is the $K$-linear transformation $f:Mat (3 \times 2, K) \rightarrow K^3$ $\begin {pmatrix} a_{1 1} & a_{2 1}\\ a_{2 1} & ...
1
vote
2answers
59 views

Using transformations and basis to find standard matrices

Let $A =\{(1,3), (2,5)\}$ be a basis of $\mathbb{R}^2$. Let $M =\left[\begin{array}{rr} 1 & -2\\ 3 & 0\end{array}\right]$ be the standard matrix for the linear transformation from ...
0
votes
1answer
55 views

Representation of Linear Transformation with respect to basis please helppp

Let $A = (1,3) (2,5)$ be a basis of $\mathbb{R}^2$. Let $M =\left[\begin{array}{rr} 1 & -2\\ 3 & 0\end{array}\right]$ be the standard matrix for the linear transformation from $\mathbb{R}^2$ ...
1
vote
1answer
29 views

How do I determine a vector x $\in \mathbb R^4$ that satisfies $T(x) = A$

$T$ is a linear transformation $A = \begin{bmatrix} 2\\4\\8\\ \end{bmatrix}$ How can you define a vector $ x \in \mathbb R^4 $ which satisfies $T(x) = A$ this is just a made up example. Forgive me ...
0
votes
1answer
53 views

Linear transformations… $T(\mathbf v)\ne0$, $T^2 - 0$… Prove $\mathbf v$ and $T(\mathbf v)$ are linearlly independent…

Suppose $T:\mathbb R^n \to\mathbb R^n$ is a linear transformation and suppose that $\mathbf v$ is a vector such that $T(\mathbf v) \ne 0$ but $T^2(\mathbf v) = 0$ (where $T^2 = T \circ T$) Prove ...
0
votes
2answers
41 views

Find the matrix of $T$ given :$ T([3, 5]) = [2, -1]$ and $T([1, 2]) = [3, 7]$ , $T$ is linear.

I know that T(v) = v' = Av , where v is the vector, v' is the image, and A is the matrix of transformation So I've set the two images (v') equal to the matrix [S sub1, S sub 2] What does [S sub1, S ...
1
vote
4answers
72 views

Proving the standard matrix U of T to be orthogonal

So my class is getting into orthogonality, however, our reading assignments haven't been touching on transformations. I have this proof problem that I cannot seem to get around. Does anyone have any ...