Linear Maps (Transformations) - Linear Algebra
11
23 hrs
$ 10.00
MTH 202: Mathematical Methods IIComprehensive treatise of advanced mathematics covering vector calculus, complex numbers, linear vector spaces, linear maps, matrices, eigenvalues and eigenvectors.
Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.
Comprehensive treatise of advanced mathematics covering vector calculus, complex numbers, linear vector spaces, linear maps, matrices, eigenvalues and eigenvectors. Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.
Course Chapters
1Introduction
Welcome and course outline. Review of elementary matrix and linear vector space concepts.
Chapter lessons
2Maps
Definition of a map (also mapping or transformation) between two sets, notations and some general properties of maps.
Chapter lessons
3.Image15:54
3Linear Maps
Definition, examples and general properties of linear maps.
Chapter lessons
14.Worked examples (11)11:07
More worked examples on proof of linearity of maps.
15.Worked examples (12)6:36
More worked examples on proof of linearity of maps.
16.Worked examples (13)21:44
More worked examples on proof of linearity of maps.
17.Worked examples (14)10:18
More worked examples on proof of linearity of maps.
18.Worked examples (15)16:19
More worked examples on proof of linearity of maps.
4Algebra of Linear Maps
Analysis of the vector space of all linear maps between two given vector spaces. Addition, scalar multiplication, composition and inverse of linear maps.
Chapter lessons
1.Addition17:00
Addition of linear maps and its linearity.
2.Scalar multiplication13:29
Scalar multiplication of linear maps and its linearity.
3.Vector space of linear maps1:07:03
Vector space of all linear maps between two vector spaces over the same field of scalars.
4.Composition18:52
Composition of linear maps - definition and proof of its linearity.
5.Properties15:43
Properties of the composition or product of linear maps - associativity, identity and distributivity.
6.Worked examples (1)9:59
Worked examples on composition of linear maps.
7.Worked examples (2)12:00
More worked examples on composition of linear maps.
5Operations on Linear Maps
Further operations on linear maps involving their definition and properties.
Chapter lessons
1.Illustration7:04
Solving problems involving derivation of the definition of a linear map from some known images.
2.Worked examples (1)26:38
Worked examples on problems involving derivation of the definition of a linear map from some known images.
3.Worked examples (2)6:49
More worked examples on problems involving derivation of the definition of a linear map from some known images.
4.Worked examples (3)27:19
More worked examples on problems involving derivation of the definition of a linear map from some known images.
6Kernel
Meaning of the kernel of a linear map, with worked examples.
Chapter lessons
1.Definition23:28
Meaning and illustration of the kernel of a linear map.
2.Worked examples (1)19:58
Worked examples on kernels of linear maps.
3.Worked examples (2)17:30
More worked examples on kernels of linear maps.
4.Worked examples (3)15:32
More worked examples on kernels of linear maps.
7Range
Meaning of the range of a linear map, with worked examples.
Chapter lessons
1.Definition24:54
Meaning and illustration of the image of a linear map.
2.Worked examples (1)37:35
Worked examples on images of linear maps.
3.Worked examples (2)24:22
More worked examples on images of linear maps.
4.Worked examples (3)9:48
More worked examples on images of linear maps.
8Matrix Representation
Matrix representation of a linear map, with worked examples.
Chapter lessons
1.Theorem32:26
Existence of matrix representation of a linear map.
2.Proof37:34
Proof of the existence of matrix representation of a linear map.
3.Procedure7:50
How to find the matrix representation of a linear map with respect to given bases of the domain and co-domain.
4.Worked examples (1)19:22
Worked examples on matrix representations of linear maps.
5.Worked examples (2)22:21
More worked examples on matrix representations of linear maps.
6.Worked examples (3)17:23
More worked examples on matrix representations of linear maps.
9Transition Matrix
Meaning and properties of the transition matrix between two bases of a vector space.
Chapter lessons
1.Definition25:36
Meaning of the transition matrix between two bases of a vector space.
2.Change of bases (1)23:55
How a change of the domain basis affects the matrix representation of a linear map.
3.Change of bases (2)16:55
How a change of the co-domain basis affects the matrix representation of a linear map.
4.Change of bases (3)18:48
How a change of bases of both the domain and co-domain affects the matrix representation of a linear map.
5.Worked examples (1)11:00
Worked examples on the transition matrix between two bases of a vector space.
6.Worked examples (2)46:00
More worked examples on the transition matrix between two bases of a vector space.