Tagged Questions
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?
$$\begin{pmatrix} 1 & ...
1
vote
1answer
37 views
Special linear transformations
Special linear transformations are matrices with determinant equal to 1.
What additional properties do such transformations have compared to "regular" linear transformations?
0
votes
1answer
29 views
Relationship between three matrices
I think this might be an odd question, and a little vague. But here goes.
This is related to coordinate transformations. Three matrices are given: $G_1 , G_2$, and $\Lambda$. $G_1$ and $G_2$ are ...
0
votes
1answer
34 views
Result of multiplying a scaling matrix with a rotation matrix
I don't understand why if you multiply a scaling matrix with rotation matrix that the resulting matrix, when applied to a shape like an ellipse, only gets scaled and does not get rotated.
$$\left( ...
4
votes
1answer
41 views
How to solve cross-products including matrices?
I'm a programmer and I'm doing a whitebalance-transformation in RGB colorspace. This should work with this transformation matrix that I've found in literature:
$$
\begin{pmatrix}
R \\
G \\
B
...
1
vote
0answers
13 views
Convert hermitian matrix to symmetric
Is there some simple transformation (or a simple way to find it) which would convert any given hermitian matrix $A$ to a symmetric matrix $B$ with the same spectrum as that of $A$ (so I guess that ...
0
votes
0answers
27 views
Convert between View Matrix and Tuple of Camera Position, LookAt Vector, Up-Vector
given a View-Matrix $M$ that can transform world coordinates into camera space, how can I convert between this representation and a more human readable form of Position ($\vec p$), Look-at vector ...
1
vote
2answers
79 views
Find the standard matrix of the transformation $T:\mathbb{R}^2\to \mathbb{R}^2$ that corresponds to the reflection through the line
Find the standard matrix of the transformation $T:\mathbb{R}^2\to \mathbb{R}^2$ that corresponds to the reflection through the line $x_2=2x_1$ followed by reflection through the line $x_1=3x_2$
I am ...
-4
votes
0answers
127 views
$[T]_{\scr{C}}=[T]_{\scr{B}}^{\scr{C}}[v]_{\scr{B}}$: A Desideratum for Creative Disquisition [closed]
$\blacklozenge\hspace{.2cm}$Consider the following data in this first litany:
$\hspace{2cm}\Diamond\hspace{.5cm}$$V$ and $W$ are finite-dimensional vector spaces.
...
1
vote
1answer
62 views
extract [0 … 2PI] rotation from 3x3 homogeneous 2D transformation matrix
I'm trying to extract the angle [0 ... 2PI] from a 3x3 homogeneous 2D transformation matrix. In fact I've found 2 very helpful posts at stackoverflow and stackexchange.
...
1
vote
0answers
21 views
Linear 2D transform in the sense of geometric figures?
Consider tranformation which turns one aligned rectangle to another:
This tranformation can be written in matrix form in the following way
where
...
0
votes
1answer
73 views
Construct a matrix transformation, majority of work done need confirmation [duplicate]
Consider
$\frac{dx}{dt} = Ax$ where $A$ is the matrix
$$
\begin{bmatrix}
1 & 0 & 1 \\
0 & 0 & -2 \\
0 & 1 & 0 \\
\end{bmatrix}
$$
...
1
vote
1answer
44 views
Local axis follows origin node rotation
I'd like to define a local axis (unit vectors l, m and n) which once defined follow the rotation of the origin node, i.e. regardless of the deformation the local axis should be basically the same as ...
0
votes
0answers
34 views
Calculation Projectionmatrix of a Camera by XYZ - UV Pairs
i have broken down my problem to plainmath and could really use some help.
Basis: I have an image. In this image I have several UV-XYZ pairs. So i know the 3d position of serveral Pixels.
Given the ...
1
vote
1answer
86 views
2
votes
1answer
75 views
Transforming matrix-equation to overdetermined minimum problem
i have broken down my problem to plainmath and could really use some help.
Basis: I have an image. In this image I have several UV-XYZ pairs. So i know the 3d position of serveral Pixels.
Given the ...
0
votes
0answers
139 views
Linear Transformation Question (Finding Standard Matrix and Image) [closed]
Please Help! here is the question:
Let T be a linear transformation from R^3 to R^4 with
T(e_1)=(<6>,<4>,<5>,<0>) and T(e_2)=(<-2>,<3>,<2>,<2>) and T(e_3)= ...
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. ...
1
vote
2answers
53 views
Basic questions regarding matrix algebra.
I had two true/false questions on my exam of which I missed.
$1)$ The map $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by $T(x)=x+e_1$ is a linear transformation.
I know this to be false, because ...
3
votes
2answers
82 views
How to compute a matrix for rotating and centering rectangle in viewport?
I have a rectangle given by 4 points. I'm trying to compute a transformation matrix such that the rectangle will appear straight and centered within my viewport.
I'm not even sure where to begin.
...
1
vote
1answer
132 views
How to find a transformation matrix having several original points and their respective transformed results?
I have three original points $pt_1, pt_2, pt_3$ which if transformed by an unknown matrix $M$ turn into points $gd_1, gd_2, gd_3$ respectively. How can I find the matrix $M$ (all points are in ...
0
votes
1answer
44 views
Transpose of 2 matrices together
So if I have an $m\times n$ matrix $A$ and I represent that matrix as $\displaystyle A = QR$, how do I write $A^{T}$ (transpose) in terms of the original $\displaystyle QR$? Does it become ...
0
votes
1answer
21 views
How to transform this matrix & swap its columns?
I'm looking for a transformation matrix (or set of transformation matrices) that transforms matrix $\mathbf A = \begin{pmatrix} a&b&i&j\\ c&d&k&l \\ e&f&m&n \\ ...
3
votes
1answer
59 views
How to “flip” and change the sign of one particular row of this matrix?
I would like to transform the following matrix :
$\mathbf A$ =$\ \begin{bmatrix}
a&b\\
c&d\\
e&f\\
g&h
\end{bmatrix}\ $ into this one : $\mathbf B$ = $\ \begin{bmatrix}
g&-h\\
...
0
votes
2answers
39 views
An explanation about terminology in vector spaces
Call a linear transformation $\rho: V \to V$ ($V$ is a vector space) idempotent if $\rho^2 = \rho$. Prove that if $\rho$ is idempotent, then it acts as the identity on $\rho(V)$.
If I understand the ...
2
votes
2answers
64 views
Meaning of $p(\phi)$ where $\phi (x,y) = (x+y, x- 2y)$ and $p(x) = x^2 -2x + 1$
Consider the linear transformation $\phi : \mathbb{R}^2 \to \mathbb{R}^2$ defined by $\phi (x,y) = (x+y, x- 2y)$. Let $p(x) = x^2 -2x + 1$. Does $p(\phi)$ make sense and if yes what is it?
3
votes
3answers
216 views
Find the spanning set of the range of the linear transformation $T(x)=Ax$.
Let
$$
A=
\begin{bmatrix}
-4 & -4 & 12 & 0 \\
-4 & -4 & 12 & 0 \\
4 & -2 & 0 &-6 \\
1 &-4 &7 &-5 \\
...
1
vote
1answer
55 views
Householder reflections
Let $x=\begin{bmatrix}
1\\
2\\
3
\end{bmatrix}$
I want to use a Householder reflector U to keep only first element in vector x, and make everything else zero
but I'm doing something wrong...
...
2
votes
3answers
131 views
Rank of matrix AB when A and B have full rank
Define $A$ as $m\times n$ matrix with rank $n$, and $B$ as $n\times p$ matrix with rank $p$. Calculate the rank of matrix $C=AB$.
--edit--
Rank of a matrix is the number of linear independent rows.
4
votes
1answer
99 views
If $AB=0$, then $A+A^T$ or $B+B^T$ is singular
Define $A$ and $B$ as being square matrices of dimension $2011$. Prove that if $AB=0$, then at least one of matrices $A+A^{T}$ or $B+B^{T}$ have rank below $2011$.
-- edit --
Rank of a matrix is ...
1
vote
0answers
114 views
Multiple integral variable substitution using Jacobian matrix and matrix rotations
Question: By an appropriate choice of new variables evaluate the integral ${\int\int}_R(x^2+y^2)dxdy$ over the interior of the square bounded by $y=\pm x$ and $y=\pm (x-2)$.
I sketched the square ...
2
votes
1answer
145 views
Reconstructing a Matrix in $\Bbb{R}^3$ space with $3$ eigenvalues, from matrices in $\Bbb{R}^2$
I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three ...
1
vote
1answer
440 views
Find the Matrix A of the Linear Transformation
Can anyone walk me through the steps to complete this problem? I am unsure of where to start to solve the problem. I get that the resulting matrix A should be a $2 \times2$ matrix, should I be finding ...
1
vote
1answer
100 views
Find the Matrix of a Linear Transformation.
It's been a few weeks since the subject was covered in my Linear Algebra class, and unfortunately linear transformations are my weak spot, so could anyone explain the steps to solve this problem?
...
1
vote
1answer
66 views
Derive Rigid Transform Matrix from Axes and Origin
I'm trying to derive the matrix of a rigid transform to map between two coordinate spaces. I have the origin and the axis directions of the target coordinate space in terms of the known coordinate ...
0
votes
1answer
149 views
Subspaces, transformation matrices exercise
I have trouble understanding the following exercise so I would really appreciate any help you could give me:
Let $k$ be a non zero vector in $\mathbb R^n$, written in standard basis. Here is a ...
0
votes
3answers
665 views
Image and Kernel of a Matrix Transformation
So I had a couple of questions about a matrix problem. What I'm given is...
Consider a linear transformation $T: \mathbb R^5 \to \mathbb R^4$ defined by $T( \overrightarrow{x} )=A\overrightarrow{x}$, ...
1
vote
1answer
86 views
What is the Matrix corresponding to a Linear Transformation
Given $T: P_2 \rightarrow P_3$ defined by:
$T(at^2 + bt +c) = (a-b+c)t^3 + (-a + 3b - 2c)t^2 +(-a-b)t +(2b-c)$
What is the corresponding Matrix of $T$?
This is what I have: First I rewrite the ...
2
votes
1answer
139 views
Linear Algebra Question ( rank of matrix )
Let $\bf A$ be an $m \times n$ matrix. If $\bf P$ and $\bf Q$ are invertible $m \times m$ and $n \times n$ matrices, respectively
prove $\operatorname{rank}(\mathbf{PA}) = ...
0
votes
0answers
41 views
Convert point coordinates
I have to create some transformations for a 3D application but I'm not very good at math.
I have 2 objects in space (let's call them ...
0
votes
1answer
85 views
Finding reflection of a matrix
To 2 decimal places, what is the value of the lower-right entry in the reflection matrix $Q_a $if a = 1.05?
Not even sure where to begin, is there a formula?
This is what I could find in my textbook
...
0
votes
2answers
110 views
3D transformation between two polylines problem
Say I have 2 separate objects.
One is a line defined by two points, the other is a polyline defined by three points.
Line 1 consists of the set of two points:
$a=(0,0,0)$ and b=$(0,0,1)$
Line 2 ...
2
votes
2answers
121 views
Transformation matrix to go from one vector to another
I've two vectors $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$. How to find transformation matrix for transform from a to b?
2
votes
0answers
111 views
transformation of coordinate systems by rotation
I am trying to convert a set of coordinates from ECEF (Earth Center Earth Fixed) to ENU (East North Up). The operation is performed by applying a rotation matrix as shown in:
...
4
votes
1answer
68 views
Can a transformation matrix be expressed in terms of the vector to be transformed?
I'm currently learning linear algebra with my friend via an online course, and we have a disagreement that we would like settled.
Upon learning that vectors can be projected onto lines by a simple ...
1
vote
1answer
75 views
determinant of matrix of transformation from Cartesian to orthogonal curvilinear
Let $(x_1, x_2)$ and $(y_1, y_1)$ be two orthogonal coordinate system with unit vectos $(\hat i_1, \hat i_2)$ and $(\hat e_1, \hat e_2)$ respectively defined by the
$x_1 = x_1(y_1,y_2)$ and $x_2 = ...
0
votes
0answers
221 views
rotation and converting between coordinate systems
Let's say you have coordinates 'a','b','c','d' in the coordinate system (x,y), and it needs to be transformed to coordinates 'e','f','g','h' in the coordinate system (x',y').
Example
...
0
votes
2answers
266 views
convert values from one coordinate system (x,y) to another coordinate system (x', y')
Following is a graph that contains both coordinate systems (x,y) and (x',y'). x, y, x', and y' are all axes
...
1
vote
1answer
281 views
Given a matrix, find a linear transformation that uses it
The matrix is:
$$\begin{pmatrix}
3+l & 8 & 3 & 3+l \\
8 & 9 & 3 & 7 \\
3 & 3 & 7 & 8 \\
3+l & 7 & 8 & 13 \end{pmatrix}$$
I'm given the above ...
1
vote
0answers
138 views
2D Cartesian Matrix / coordinate transformation.
I has initially asked this question in the programming site but did not get an answer that worked. This is my first question on this site so please bear with me.
Consider a page with three distinct ...
