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:
- T(ku) = kT(u) (Homogeneity property)
- T(u+v) = T(u)+T(v) (Additivity property)
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
- T is said to be one-to-one if T maps distinct vectors in V into distinct vectors in W
- T is one-to-one if and only if ker〖(T)={0}〗
- 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
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