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# Linear Transformatio

### Linear Transformation, Kernel, Range, Eigenvalue/Vector, diagonalization

Linear Transformation Let V & W be two vectors spaces. A map T frm V to W.
Linear Transformation from Rn to Rm is of the form ... T(x) = Ax with A= [T(e1):T(e2):...:T(en)]
Ck(I) Space of continuous function on the I interval
Mapping T:(V -W) with V and W be vector spaces T from V onto W is a rule that assigns to each vector v in V precisely one vector w=T(v).
Linearity properties T(u+v) = T(u) = T(v) for all u,v in V T(cv) = cT(v) for all v in V and all scalars c.
A mapping T:V-W is called ilnear transforamtion from V to W if .... Linear Properties are true. The vector space V is called the domain of T, whie the vector space W is called the codomain of T.
Let T:V-W be a linear transformation. Then ... T(0v) = 0w T(-v) = -T(v) for all v in V.
A linear transformation T:Rn - Rm defined by T(x) = Ax is an m x n matrix called ...; Matrix Transformation
Ker (T) = {v in V: Tv = 0}
Ran ( T) = (or Rng(T)) {T(v) of W : v in V}
Kernel Let T:V-W be a linear transforamtion.the set of all vector v in V such that T(v)=0
Range A linear transformation T:V-W , the subset of W consisting of all transformed vector from V.
If T:Rn-Rm is the linear transformation with matrix A, then ... Ket(T) is the solution set to the homogeneous linear system Ax=0.
If T:Rn - Rm with m x n matrix A , then Ket(T) = nullspace(A) subspace of Rn
If T:Rn - Rm with m x n matrix A , then Ran(T) = colspace (A) subspace of Rm
If T:V-W is a linear transformation, then Ker(T) = is a subspace of V
If T:V-W is a linear transformation, then Ran(T) = subspace of W
General Rank-nullity theorem dim[Ran(T)] + dim[Ker(T)] = dim[V]
Created by: DrMolina