A Brief Summary of Linear Transformation [LA3]
In this series of posts, I`ll be writing about some basics of Linear Algebra [LA] so we can learn together. The book I`ll be using as the material is:
Cabral, M. & Goldfeld, P. (2012). Curso de Álgebra Linear: Fundamentos e Aplicações. Third Edition.
Last time we talked about Linear Systems and Matrices. Now, it’s time to know more about linear transformation.
In the context of linear algebra, we use linear transformation as a synonym for a function.
Matrices that represent geometrical transformations like rotation and reflection are used in computer graphics and for the study of linear transformations.
Linear Transformation and Matrix
Being V and W vector spaces. A function (or transformation) T: V → W is called a linear transformation if:
- T(ku) = kT(u)
- T(u + v) = T(u) + T(v)
For all u,v ∈ V and scalar k.
Examples of linear transformations:
a) A linear transformation is called identity if there is a transformation I: V → V defined by I(v) = v for all v ∈ V.
b) 0: V → W is defined by 0(v) = 0 for all v ∈ V.
c) T: ℝ³ → ℝ² defined by T(x, y, z) = (z, -x)
A matrix can be associated with a linear transformation. That is, given a matrix A m x n, we define T_A: ℝ^n → ℝ^m by T_A(w) = Aw.
Being T: U → V a linear transformation and {u1, u2, …, un} a basis of U. If we know T(ui) for i = 1, …, n, then T(u) is well determined for any u ∈ U.
Example: Determine a matrix associated with a linear transformation
a) T(x, y, z, w) = (x - y + 2z, x + y, z +w)
Substitute the canonical basis {(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)} in the linear transformation given. That means that T(1,0,0,0) = (1–0+2*0, 1 + 0, 0 + 0) = (1,1,0). The same applies to the other canonical basis: T(0,1,0,0) = (-1,1,0); T(0,0,1,0) = (2,0,1); T(0,0,0,1) = (0,0,1). After that, put those vectors as columns of the matrix T:
Kernel and Image
A kernel of a linear transformation T: U → V, or Ker(T), is a set of the vectors in the domain where the image is a null vector:
Ker(T) = {u ∈ U | T(u) = 0}
Nullity of a linear transformation T is the dimension of its kernel: dim(Ker(T))
An image of a linear transformation T: U → V, denoted by Im(T), is the set of vectors of the Codomain that are an image of some vector in the domain:
Im(T) = {v ∈ V | v = T(u) for some u ∈ U}
The rank of a linear transformation T is the dimension of its image: dim(Im(T))
Rank-Nullity Theorem
Being T: U → V linear where U has a finite dimension. Then
dim(Nuc(T)) + dim(Im(T)) = dim(U)
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