3
votes
2answers
55 views

Prove that an element of the basis is an element of the Kernel after linear transformation

Let $T:R^4\rightarrow R^4$ and basis $B=(v_1,v_2,v_3,v_4)$. $$T(v_1)+T(v_2)=T(v_3)\; \text{ and } \; T(v_1)+T(v_3)=T(v_2)$$ Prove that $v_1\in Ker(T)$ What I wrote is: $$T(v_1)=T(v_3)-T(v_2)\; ...
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
1answer
26 views

Why does the discrete cosine transform compact the information at the “low frequencies”?

I've been investigating about the discrete cosine transform. I think I understand the practical applications it has and how it is used in image/audio compression. I also know it is related with the ...
1
vote
1answer
40 views

How do you prove a hilbert transform?

I am stuck with this question below, I need help;
0
votes
0answers
24 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
2answers
34 views

What does it mean when dim(V)=rankT

I have a question relating to a linear transformation and have ended up with the result that $dim(V)=rank(T)$. I got to this because I'm told that $V$ and $W$ are finite dimensional vector spaces, ...
0
votes
1answer
19 views

Transformation Matrix of a linear function

Consider the function $f: \mathbb{R}^3 \rightarrow \mathbb{R}^2$. Let $A = \{ (1,2,3)^t, (1,0,4)^t,(0,0,2)^t \}$ a base of $\mathbb{R}^3$ and $B = \{ (1,1)^t , (2,1)^t) \}$ a base of $\mathbb{R}^2$. ...
0
votes
1answer
28 views

Is a linear transformation just a mathematical description of a straight line?

On Physics Stack Exchange, the question was asked: Are lorentz transformations linear? The up-votes given to an answer seemed to be in proportion to how mathematically sophisticated it was, with mine ...
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
1answer
33 views

Finding image and the null space of a linear transformation

Take $x \in \mathbb{R}^n\backslash\{0\}$ and let $L = \text{span}\{x\}$. Now we consider the linear transformation $$T \colon \mathbb{R}^n \to \mathbb{R}^n,$$ to be given by $T(y) = \text{proj}_{L}y ...
0
votes
1answer
48 views

Why would the discrete fourier transform “see” signals like this? What is the origin of spectral leakage?

The discrete fourier transform of $x = (x_{0},\dots,x_{N-1})$ is defined as $\displaystyle X_{k} = \sum_{n=0}^{N-1} x_{n}\omega^{kn}_{N}$ where $\omega^{kn}_{N} = e^{-2\pi ik/N}$ and ...
2
votes
0answers
31 views

Given the minimal polynomial, find the largest invariant subspace

Let the linear transformation T on the vector space $V$ over $\mathbb{Q}$ have minimal polynomial $(x^{7} - x^{3})$. a) What is the largest invariant subspace W of V for which $T (W) = W$? b) Find a ...
0
votes
0answers
8 views

Stable equilibrium position of 3d models.

I have 2 models, described by vertices arrays. The aim is to find stable equilibrium position of one of the models upon the other. The algorithm should consider the possibilities of transformation of ...
1
vote
0answers
15 views

A question on the procedure of finding the matrix of a linear transformation of a polynomial and a combination of its derivatives

I'm trying to self-study Linear Algebra from Linear Algebra Done Wrong, but the book doesn't have solution manual so my question might be extremely easy, apologize in advance: The question is for the ...
1
vote
2answers
45 views

Find the matrix that represents a rotation clockwise around the origin by$ 30∘$ followed by a magnification by a factor of 4.

Find the matrix that represents a rotation clockwise around the origin by 30∘ followed by a magnification by a factor of 4. My attempt: I multiplied the magnification matrix $\left[ ...
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; ...
1
vote
2answers
24 views

Linear transformation - linear matrix & kernel

I have a problem understanding getting the KERNEL and IMAGE of a linear transformation. We have the following transformation given: $$ \mathbb{R}_{2}[ x ] \rightarrow \mathbb{R}_{2}[ x ] $$ $$ (\phi ...
0
votes
1answer
22 views

Translation of basis for a vector space on the specified distance

In the Euclidean space $XYZ$ is a basis $X_1Y_1Z_1$ defined that is specified by the vectors $\overrightarrow {O_1X_1}$, $\overrightarrow {O_1Y_1}$ and $\overrightarrow {O_1Z_1}$. How to calculate ...
0
votes
1answer
35 views

Determine whether the following map is a linear transformation.

So I have to determine if the following is a linear transformation: $$T: F(I) \rightarrow F(I)$$ defined by: $$T(f) = 2f$$ I know that if you let $T: V\rightarrow W$ be a linear transformation. Then: ...
1
vote
1answer
55 views

Find the equation of the linear transformation of an orthogonal projection on the line y=mx.

Let $T : \mathbb R^2 → \mathbb R^2$ the orthogonal projection on the line $y = mx$. Prove that for all $a, b \in \mathbb R$, $$\begin{align}T((a,b)) = {\frac{1}{m^2 + 1}}(a+mb, ma + ...
1
vote
1answer
58 views

Linear Algebra - Question about transformation and characteristic polynomial

I have some trouble with this question, I tried to solve it but I'm not sure that my solution is correct. I'll be glad if somebody could take a look. Data : T : R^4 --> R^4 (linear transformation) ...
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;
3
votes
3answers
171 views

Basis in the vector space of all polynomials

Let $V$ vector space of all polynomials $p(t) = a_0 + a_1t + \cdots + a_nt^n$,$\forall n \in\mathbb{N}$ and $a_0,\ldots,a_n \in\mathbb{R}$. How can I prove that $ \gamma = \{1,t,t^2,\ldots\}$ is a ...
2
votes
3answers
48 views

What kind of transformation an upper triangular matrix represents

Every matrix represents a linear transformation, but depending on characteristics of the matrix, the linear transformation it represents can be limited to a specific type. For example, an orthogonal ...
3
votes
2answers
31 views

Rational quadratic forms

The quadratic form $$10x^2+20y^2+2z^2+4xy-6xz+8yz$$ can be written as $x^TAx$, where A = [ [10,2,-3] , [2,20,4] , [-3,4,2] ] Using diagonalization, this can be written in the form ...
1
vote
0answers
31 views

Linear Probability Density Transformations

Suppose that $\mathbf{y=Ax}$ and that a probability density function over $\mathbf{x}$ is defined as $p(\mathbf{x})$. If $\mathbf{A}$ has an inverse then the PDF over $\mathbf{y}$ is given by ...
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
3answers
31 views

Find that the given linear transform is a isomorphism

I'm studying Linear Algebra and I'm having trouble demonstrating that a function is a isomorphism, that is: "Given the linear transform $T: V \rightarrow W$, $T$ is a isomorphism if and only if it is ...
-1
votes
1answer
29 views

Understanding a definition for vector-spaces

Let $V$, a finite dimensional vector space, and $L$, a subspace of $V$. Let $T:V^*\rightarrow L^*$ defined as: $T(\varphi)(x)=\varphi(x)$ for all $\varphi \in V^*$. Prove $T$ is onto. Well, I'm ...
3
votes
3answers
70 views

$V = \operatorname{Im} T + \ker T $ then $ \operatorname{Im} T \cap \ker T = \{0\}$

Let $F$ be a field, let $V$ be a vector space with finite dimension over $F$ and let $T$ be a linear operator on $V$. Prove that: a) If $V = \operatorname{Im} T + \ker T $ then $\operatorname{Im} T ...
1
vote
1answer
29 views

$\ker S$ is not contained in $\ker T$ implies $\dim \Im T \ge 1$

Let $T,S:V\rightarrow W$.where $V$ is a finite vector space above $F$ and $W$ is one-dimensional vector-space above $F$ ($\dim W = 1$). It is given that $\ker S$ isn't contained in $\ker T$. Why is ...
1
vote
0answers
39 views

what's a homogeneous transformation?

Minkowski writes in his paper on Time and Space: If, for simplicity, we retain the same zero point of space and time, the first-mentioned group signifies in mechanics that we may subject the axes ...
2
votes
1answer
57 views

Isomorphism implies direct sum of Kernel and Image

If $f: U \rightarrow V$ and $g: V \rightarrow W$ are linear transformations between vector spaces over a field $K$ such that $ g \circ f$ is an isomorphism, then $V = \operatorname{Im}f \oplus ...
1
vote
0answers
40 views

$T (x_1,x_2,x_3,…,x_n) = (-x_3,x_3,x_4,x_5,…) $ then $ W \ne ker T$

Let $V$ the vector space of all sequences of real numbers and $W$ the subspace given by $W = \{(a,a,0,0,...) | a \in R\}$ , and $T : V \rightarrow V$ given by $T (x_1,x_2,x_3,...,x_n) = ...
1
vote
2answers
29 views

If $f\in V$ of degree $n$ then for every $g \in P_n(\Bbb R)$ there exist scalars $c_0,c_1,..,c_n$ such that $g = c_0f + c_1f'+ … + c_nf^{(n)}$

Let $V=P(\Bbb R)$ and $1 ≤ i$ be the vector space of the polynomials with real coefficients, on the field of real numbers $\Bbb R$. Let $T_i(f)=f^{(i)}$ the $i$th derivate of $f$. a) I have to show ...
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 + ...
1
vote
2answers
20 views

Showing a set is a basis

Let $V, W$ vector spaces and $f, g:V\rightarrow W$, linear transformations. $\ker f \subset \ker g$. Now, let $\{v_1,...,v_n\}$ a basis for $\ker f$ and we'll complete it with $\{u_1,...,u_m\}$ to a ...
1
vote
4answers
87 views

$\{ v_1,v_2,…,v_n\}$ is basis of $V$ if and only if $\{ v_1,v_1 + v_2,…,v_1 + v_2+…+v_n,\}$ is a basis of $V$

Let $V$ a vector space over a field $K$. Is it true $\{ v_1,v_2,...,v_n\}$ is basis of $V$ if and only if $\{ v_1,v_1 + v_2,...,v_1 + v_2+...+v_n,\}$ is a basis of $V$ ? I made some examples and ...
0
votes
1answer
39 views

$V$ and $W$ finite vector spaces with dimension $n$ and $r$ with $\{ v_1,v_2,…,v_n \} \subset Ker T$ and $\{ u_1,u_2,…,u_s \} \subset V$

Let $V$ and $W$ finite vector spaces with dimension $n$ and $r$ respectively and $T: V \rightarrow W$ linear transformation, $\{ v_1,v_2,...,v_n \} \subset Ker T$ and $\{ u_1,u_2,...,u_s \} \subset ...
1
vote
3answers
52 views

Diagonalizable operator of a finite vector space

Let $V$ a vector space of finite dimension, $dim (V) = r$, and $T: V \rightarrow V$ a diagonalizable operator with $ \lambda _1,\lambda_ 2,...,\lambda _r$ distincts eigenvalues of $T$ then $ (T- ...
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
4answers
68 views

Showing some transformation is linear

Let $T: P_3(\mathbb{R}) \to P_3(\mathbb{R})$ be an operation defined by $$T(a+bx+cx^2+dx^3) = a + dx + (a+d)x^2 +(b-c)x^3$$ Show that $T$ is linear What I have done so far is look at it like ...
1
vote
2answers
48 views

$\dim(V) = \dim T(V) + \dim T^{-1}(0)$

Let $T\colon V \rightarrow W$ a linear transformation between the real vector spaces $V$ and $W$ both with finite dimension. How can i prove that $\dim(V) = \dim T(V) + \dim T^{-1}(0)$. I can't ...
0
votes
1answer
36 views

Can someone help ? I have this answer of linear transformation.

Onto? What we have in class that if $n=2$ and $m=3$ that clear $2<3$ ,$T$ will be not onto. He said make three point u have with three variable $=(y,z)$ .then u will have one free variable that ...
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?
0
votes
1answer
30 views

Randomly generating special affine transformations

I want to generate many random special affine transformations, that is, affine transformations that preserve volume (determinant equal to 1). I need quite a few of them. Is there a better way than ...
0
votes
0answers
27 views

change of a basis

lets say i have following bases: $C=\left\{\begin{bmatrix}1\\ 1\end{bmatrix},\begin{bmatrix}1\\ -1\end{bmatrix}\right\}$ $S=\left\{\begin{bmatrix}1\\ 0\end{bmatrix},\begin{bmatrix}0\\ ...
1
vote
3answers
38 views

Find $T(x,y,z)$ for $(x,y,z)$ in $\mathbb R^3$

I have a question with the following facts: $T: \mathbb R^3\to\mathbb R^3$ is a linear transformation represented by the basis $B=((1,0,0),(1,1,0),(1,1,1))$, by the matrix $[T]_{B}$, where $[T]_{B}$ ...