Matematika Diskrit : Relasi, Sifat Relasi, dan Contoh Soal

Matriks dan Ruang Vektor : Transformasi Linier Umum

General Linear Transformations

Definition

T:V→W is a mapping from a vector space V to a vector space W 

T is called a linear transformation from V to W if the following two properties hold for all vectors u and v and for all scalars k: 

1. T(ku) = kT(u) (Homogeneity property) 
2. T(u+v) = T(u)+T(v) (Additivity property)

is T linear transformation ?

Linear Transformation from Images of Basis Vectors

T:V→W be a linear transformation, where v is finite dimensional

S = {v_1,v_2,…,v_n} is a basis for V

The image of any vector v in V can be expressed as 

T(v) = c_1 T(v_1 )+c_2 T(v_2 )+…+c_n T(v_n ) 

Where c_1,c_2,…,c_n are the coefficients required to express v as a linear combination of the vectors in the basis S

Example : 

Consider the basis S={v_1,v_2,v_3} for R^3, where 

v_1 = (1,1,1), v_2 = (1,1,0), v_3 = (1,0,0)

Let T:(R^3→R^2) be the linear transformation for which 

T(v_1 ) = (1,0), T(v_2 ) = (2,-1), T(v_3 ) = (4,3)

 

Find a formula for T(x_1,x_2,x_3) and then use that formula to compute T(2,-3,5)

Kernel and Range

 T:V→W is a linear transformation 

ker(T): 

• kernel of T is the set of vectors in V that T maps into 0 
• Subspace of V 
• If kernel of T is finite-dimensional, its dimension is called nullity of T 

R(T): 

• range of T is the set of all vectors in W that are images under T of at least one vector in V 
• Subspace of W 
• If the range of T is finite-dimensional, its dimension is called the rank of T 

Theorem: rank(T)+nullity(T) = dim⁡〖(V)〗

find ker(T), R(T), rank(T), nullity(T)!

Compositions and Inverse Transformations

One-to-One and Onto

T:V→W is a linear transformation from a vector space V to a vector space W 

1. T is said to be one-to-one if T maps distinct vectors in V into distinct vectors in W 
2. T is one-to-one if and only if ker⁡〖(T)={0}〗 
3. T is said to be onto if every vector in W is the image of at least one vector in V

Example: The linear transformation T:M_22→R^4 defined by

Show that T are both one-to-one and onto!

Composition of Linear Transformations

If T_1:U→V and T_2:V→W are linear transformations, then the composition of T_2 with T_1, denoted by T_2∘T_1 is the function defined by 

(T_2∘T_1 )(u)=T_2 (T_1 (u)) 

Where u is a vector in U

Inverse Linear Transformations

T:V→W is a one-to-one and onto linear transformation with range R(T). w is any vector in R(T). there is T^(-1) (called the inverse of T) that is defined on the range of T and maps w back into v 

If T_1:U→V and T_2:V→W are one-to-one linear transformations, then: 

• T_2∘T_1 is one-to-one 
• (T_2∘T_1 )^(-1)=T_1^(-1)∘T_2^(-1)

Example:

Let T:R^3→R^3 be the linear operator defined by the formula T(x_1,x_2,x_3 )=(3x_1+x_2, -2x_1-4x_2+3x_3,5x_1+4x_2-2x_3). 

Find T^(-1)!

Matrix Transformation

T_A:R^n→R^m can be represented by multiplication an m×n matrix A 

For Example: 

Let T:R^3→R^3 be the linear operator defined by the formula T(x_1,x_2,x_3 )=(3x_1+x_2, -2x_1-4x_2+3x_3,5x_1+4x_2-2x_3). 

Represent T by a multiplication by a matrix

T_A is not one-to-one if m<n . T_A is not onto if n<m. If m=n, T_A is both one-to-one and onto if and only if A is invertible

For Example: 

Let T:R^3→R^3 be the linear operator defined by the formula T(x_1,x_2,x_3 )=(3x_1+x_2, -2x_1-4x_2+3x_3,5x_1+4x_2-2x_3). 

• Determine whether T is one-to-one! 
• Determine whether T is onto! 
• Determine whether T is invertible? Determine T^(-1) if T invertible! 

Sumber 

Slide MRV : Transformasi Linier Umum

http://informatika.unpar.ac.id/