0
votes
0answers
26 views

linear algebra question related to basis, kernel and linear transformation [duplicate]

Let V be a 2-dimensional vector space, and let α=e1,e2 be a basis for V. Define a linear transformation T:V→V by declaring that: T(e1+e2)=2e1−e2 T(e2)=4e1−2e2. a. Find [T]α,α. (one alpha is upper ...
1
vote
1answer
46 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
126 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
36 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
30 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
49 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
30 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
62 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
20 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
36 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
46 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
36 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
28 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
34 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
18 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
80 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
38 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
48 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
38 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
84 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
67 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
44 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
47 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
0answers
17 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
26 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
37 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 $[T]_{B}$ ...
0
votes
1answer
25 views

Finding if a linear transformation exist [duplicate]

I have to following question which I find hard to solve. Is there a linear transformation from $\mathbb{R}^5$ to $\mathbb{R}^4$ such that: $T:\mathbb{R}^5→\mathbb{R}^4$ Such that: ...
0
votes
3answers
49 views

find the vector $(x,y,z) \in \mathbb{R}^3$ and the constants $\lambda \in \mathbb{R} $ such that $T(x,y,z) = (\lambda x, \lambda y, \lambda z )$

Let $T : \mathbb{R}^3 \rightarrow \mathbb{R}^3$ defined by : $$T(x,y,z) = (x-y+4z,3x+2y-z,2x+y-z)$$ How can i find the vector $(x,y,z) \in \mathbb{R}^3$ and the constants $\lambda \in \mathbb{R}$ ...
0
votes
1answer
23 views

Endomorphisms and Invariant Subspaces

I have a question or two regarding the following exercise: Let $\alpha$ be the endomorphism of $\Bbb{Q}^4$ defined by: $$\alpha : \left[\begin{matrix}a \\ b \\ c \\ d \end{matrix}\right] \mapsto ...
0
votes
1answer
18 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 ...
1
vote
3answers
32 views

Query regarding Linear Transformation…

As we always read in Complex Analysis, Linear Transformation (L.T.) is a combination of Translation, Rotation and Magnification i.e. $T(z)=az+b$ is a L.T. in complex. However, It doesn't satisfy the ...
0
votes
0answers
19 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
13 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
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 = ...
0
votes
2answers
36 views

Transformation of the points on a plane

How do I transform a point $(x,y,z)$ on plane $\Pi (ax + by + cz = 0)$ to a point $(x',y',z')$ on plane $\Phi(ax+by+cz+d=0)$? What matrix should I use? Here is a 2-D representation of what I'm ...
1
vote
1answer
45 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
32 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
48 views

matrix of orthogonal projection with respect to the ordered basis.

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
39 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 ...
1
vote
1answer
35 views

Problem with linear transformation from $\mathbb{R}^2$ to $M_{2\times 2}$

I'm trying to solve this problem, but at the end I find something's wrong with my work. Here is the problem: We're given the bases: $$ \beta = \bigg\{\begin{pmatrix}1\\1\end{pmatrix} ...
0
votes
1answer
67 views

If $x$ belongs $V$ then $Tw = w$ if and only if $x = v + k$ with $k$ in $\operatorname{ker}(T)$ [closed]

Let $F$ a field, $V$ and $W$ vector spaces, $T$ a linear transformation from $V$ to $W$, if $w$ belongs $W$ and $v$ in $V$ such that $Tv = w$, if $x$ belongs $V$ then $Tx = w$ if and only if $x = v + ...
0
votes
1answer
38 views

Compute the transformation in the given Basis

I forgot how to compute the transformation in a given basis. :'( For example, say I have the transformation \begin{equation}(a, b) \mapsto \begin{bmatrix}10a - 6b \\ 17b - 10b ...
0
votes
1answer
29 views

The linear map $ T: \mathbb R^3{\rightarrow} \mathbb R^3$ with given matrix is a rotation about some line. Find the line.

Finals studying continued. $ T: \mathbb R^3{\rightarrow} \mathbb R^3$ with matrix $$A= \begin{pmatrix} -2/7 & 6/7 & 3/7 \\ 3/7 & -2/7 & 6/7 \\ 6/7 & 3/7 & -2/7 \\ ...