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In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. (Def: )
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Conversion a 2D point value, back to 3D given a known Y value
I am dealing with an image processing problem, and I think I am doing something wrong, mathematically. I have a homogenous point in 3D $P=(x,y,z,1)$, and a corresponding homogenous point on the image ... linear-algebra linear-transformations image-processing- 1,019
Connection between kernels of linear maps of semimodules and injectivity
Let $S$ be a semiring (i.e. satisfies all the ring axioms besides existence of additive inverses) and $M, N$ semimodules over $S$ (same thing). For a linear map $\varphi : M \rightarrow N$, we can ... linear-transformations modules semiring- 620
Verify that $T(f):=\int_{-1}^0 f(x)dx-\int_{0}^1 f(x)dx$ is a bounded functional$ and find its norm
Let ($C[-1,1], \lVert \cdot \rVert _\infty)$ be a space of continuous functions on $[-1,1]$ with sup norm. Verify that $T(f):=\int_{-1}^0 f(x)dx-\int_{0}^1 f(x)dx$ is a functional and determinate if ... functional-analysis linear-transformations- 11
Eigenvectors of linear transformations: Reflections vs Rotations
I'm curious why reflections can have real eigenvectors/eigenvalues whereas rotations always have imaginary numbers. The two linear transformations seem similar to me in spirit so this difference is ... eigenvalues-eigenvectors linear-transformations rotations reflection- 183
Finding the basis of an image of linear transformation
Let $\vec{a}, \vec{b} \in \mathbb{R^3}$. Let $A : \mathbb{R^3} \rightarrow \mathbb{R^3}$ be a linear transformation and $A\vec{x} = \langle \vec{x}, \vec{a} \rangle \vec{b} + 2 \langle \vec{x},\vec{b} ... linear-algebra linear-transformations- 73
linear transformation of a line problem
I am studying maths as a hobby and have tried the following question. Find the equation of the line $\frac{x+3}{-2} = \frac{y-1}{4}$ after it has been transformed using $\begin{pmatrix} 3 & -1\\ 0 ... matrices linear-transformations- 1,149
Let $f_1, f_2 \in E^*$ such that $\{f_1, f_2\}$ is linearly independent. Is the map $x \mapsto (f_1(x), f_2(x))$ injective or surjective?
Let $E$ be a topological vector space and $E^*$ its topological dual. Let $f_1, f_2 \in E^*$ such that $\{f_1, f_2\}$ is linearly independent. Clearly, $f_1 \neq 0 \neq f_2$. We define $$ F:E \to \... functional-analysis linear-transformations topological-vector-spaces- 1,271
Which of the following expressions describes the sum [closed]
Exponential question to describe the sum I am studying Calculus, this is a homework question and I am having a hard time resolving it. It's part of my Associate degree in Business Management courses. ... calculus linear-transformations- 1
Showing $A$ is diagonalizable if $A$ commutes with a projection.
Let V be a finite dimensional real or complex vector space, B the base of V and A: V -> V an endomorphism. Prove: i) $\forall b \in B$ exists a projection $P: V \rightarrow V$, such that $\... linear-algebra linear-transformations diagonalization- 3
Does this surface have rotational symmetry?
I recently started to study symmetries on surfaces, and came with the (rather simple, but not so much for me yet) problem to check symmetries on Dini's surface, parameterized by $$ x=a \cos u \sin v \\... linear-algebra linear-transformations surfaces symmetry- 381
Do we have $(Ax,Ay) = (x,y)$ on $\mathbb{R}^n$ with any inner product when $A$ is an orthogonal matrix?
Let $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be defined by $T(\boldsymbol{x})=A \boldsymbol{x}$, where $A$ is an orthogonal matrix. Is it true that for any inner product we have $(Ax,Ay) = (x,y)$... linear-algebra linear-transformations- 109
What is the norm of $T$ : $L^1$ (($-1$,$1$) , $\mathbb{R}$) $\rightarrow$ $\mathbb{R}$
I have the following question What is the norm of $T$ : $L^1(-1,1)\to \mathbb{R}$, where $T(f)= \int_{-1}^{1} t f(t) dt$? We know that $$\|T\|= \sup_{f \ne 0} \dfrac{|T(f)|}{\|f\|_{1}}$$ I was able ... real-analysis general-topology complex-analysis linear-transformations unbounded-operators- 7
Linear Mapping Proof for Combination of Zero and Identity Maps
So I am mostly finished a proof on mapping some basis vectors and have created my own linear map, $L$, which takes vectors $v_1 \ldots v_k$ and maps them to 0 and also takes vectors $v_{k+1} \ldots ... vector-spaces linear-transformations- 67
What is the procedure to find adjoint of linear transformation on infinite dimensional inner product space?
What is the procedure to find adjoint of linear transformation on infinite dimensional inner product space? When in the procedure , we know that adjoint does not exist? linear-algebra linear-transformations- 41
Artin Algebra 4.1.5
This question comes from the 1st edition and was asked in the Harvard lecture videos on Abstract Algebra. Let $A$ be a $k\times m$ matrix and let $B$ be an $n\times p$ matrix. Prove that the rule $M\... linear-algebra abstract-algebra linear-transformations self-learning- 611
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