Essence of Linear Algebra Linear combinations and Span Some cool - PowerPoint PPT Presentation

Essence of Linear Algebra Shreedhar Kodate The Geometry of Linear Equations Vectors and Basis vectors Essence of Linear Algebra Linear combinations and Span Some cool intuitions The box game: Matrices Elimination and Multiplication, A=LU Transforming your Shreedhar Kodate 1 LIFE Leenearly Cool Video, The Determinant 1 Department of Computer Science and Automation Space Tour Indian Institute of Science, Bengaluru Column space, Null space, Inverses Celebrity: The Rank CSA Summer School, 2017 Solution concept Some things of Eigen Eigen values, Eigen vectors, Change of Basis Summary

Essence of Linear Algebra Shreedhar Kodate The Geometry of Linear Equations Vectors and Basis vectors Linear combinations and Span The box game: Matrices Disclaimer Elimination and Multiplication, A=LU All the credits for the content goes to the respective authors Transforming your LIFE Leenearly listed in the Appendix: Further Learning. Cool Video, The Determinant Space Tour Column space, Null space, Inverses Celebrity: The Rank Solution concept Some things of Eigen Eigen values, Eigen vectors, Change of Basis Summary

Essence of Linear The Storyline Algebra Shreedhar Kodate The Geometry of Linear Equations The Geometry of Vectors and Basis vectors Linear Equations Vectors and Basis Linear combinations and Span vectors Linear combinations and Span The box game: Matrices The box game: Matrices Elimination and Multiplication, A=LU Elimination and Multiplication, A=LU Transforming your LIFE Leenearly Transforming your LIFE Leenearly Cool Video, The Determinant Cool Video, The Determinant Space Tour Space Tour Column space, Null space, Inverses Column space, Null space, Inverses Celebrity: The Rank Celebrity: The Rank Solution concept Some things of Solution concept Eigen Eigen values, Eigen vectors, Change of Some things of Eigen Basis Eigen values, Eigen vectors, Change of Basis Summary

Essence of Linear The Storyline Algebra Shreedhar Kodate The Geometry of Linear Equations The Geometry of Vectors and Basis vectors Linear Equations Vectors and Basis Linear combinations and Span vectors Linear combinations and Span The box game: Matrices The box game: Matrices Elimination and Multiplication, A=LU Elimination and Multiplication, A=LU Transforming your LIFE Leenearly Transforming your LIFE Leenearly Cool Video, The Determinant Cool Video, The Determinant Space Tour Space Tour Column space, Null space, Inverses Column space, Null space, Inverses Celebrity: The Rank Celebrity: The Rank Solution concept Some things of Solution concept Eigen Eigen values, Eigen vectors, Change of Some things of Eigen Basis Eigen values, Eigen vectors, Change of Basis Summary

Essence of Linear Vectors Algebra Shreedhar Kodate What even are they? The Geometry of Linear Equations Vectors and Basis vectors Linear combinations and Span The box game: ◮ A line with a arrowhead? Matrices Elimination and OR Multiplication, A=LU Transforming your LIFE Leenearly Cool Video, The Determinant Space Tour Column space, Null space, Inverses Celebrity: The Rank Solution concept Some things of Eigen Eigen values, Eigen vectors, Change of Basis Summary

Essence of Linear Vectors Algebra Shreedhar Kodate What even are they? The Geometry of Linear Equations Vectors and Basis vectors Linear combinations and Span The box game: ◮ A line with a arrowhead? Matrices Elimination and OR Multiplication, A=LU ◮ A set of numbers arranged vertically? Transforming your LIFE Leenearly Cool Video, The Determinant   1 Space Tour Column space, Null   space, Inverses 2   Celebrity: The   Rank   3 Solution concept Some things of Eigen OR Eigen values, Eigen vectors, Change of Basis Summary

Essence of Linear Vectors Algebra Shreedhar Kodate What even are they? The Geometry of Linear Equations Vectors and Basis vectors Linear combinations and Span The box game: ◮ A line with a arrowhead? Matrices Elimination and OR Multiplication, A=LU ◮ A set of numbers arranged vertically? Transforming your LIFE Leenearly Cool Video, The Determinant   1 Space Tour Column space, Null   space, Inverses 2   Celebrity: The   Rank   3 Solution concept Some things of Eigen OR Eigen values, Eigen vectors, Change of Basis ◮ An abstract # » v Summary

Essence of Linear Vectors Algebra Shreedhar Kodate Abstract view 1 The Geometry of Linear Equations Vectors and Basis vectors Linear combinations # u + ( # » v + # » w ) = ( # » u + # » v ) + # » » w and Span The box game: Matrices Elimination and Multiplication, A=LU # v + # » w = # » w + # » » v Transforming your LIFE Leenearly There is a zero vector # 0 such that # » » 0 + # v = # » v for all # » v . For » Cool Video, The Determinant v ) = # » every vector # v there is a vector − # » v so that # » v + ( − # » » 0 . Space Tour Column space, Null space, Inverses a ( # » bv ) = ( ab ) # » 1 # v = # » » v v Celebrity: The Rank Solution concept Some things of a ( # v + # » w ) = a # » v + a # » » w Eigen Eigen values, Eigen vectors, Change of Basis ( a + b ) # v = a # » v + b # » » Summary v 1 Abstract vector spaces — Essence of Linear Algebra, Chapter 11

Essence of Linear Basis Vectors Algebra Shreedhar Kodate These are the vectors which can define the entire coordinate space. The Geometry of Linear Equations Vectors and Basis vectors Linear combinations ◮ Do you recognize this special vector? and Span The box game: Matrices   Elimination and # » i Multiplication, A=LU Transforming your   # » LIFE Leenearly  j    Cool Video, The Determinant   # » k Space Tour Column space, Null space, Inverses Celebrity: The Rank Solution concept Some things of Eigen Eigen values, Eigen vectors, Change of Basis Summary

Essence of Linear Basis Vectors Algebra Shreedhar Kodate These are the vectors which can define the entire coordinate space. The Geometry of Linear Equations Vectors and Basis vectors Linear combinations ◮ Do you recognize this special vector? and Span The box game: Matrices   Elimination and # » i Multiplication, A=LU Transforming your   # » LIFE Leenearly  j    Cool Video, The Determinant   # » k Space Tour Column space, Null space, Inverses ◮ Also, have you seen this? Celebrity: The Rank Solution concept   Some things of  1 1 Eigen  Eigen values, Eigen vectors, Change of 0 1 Basis Summary This is a special matrix called Shear matrix.

Essence of Linear Basis vectors Algebra Shreedhar Kodate The Geometry of Example Linear Equations Vectors and Basis The image 2 below shows the basis vectors of a Shear matrix. vectors Linear combinations and Span The box game: Matrices Elimination and Multiplication, A=LU Transforming your LIFE Leenearly Cool Video, The Determinant Space Tour Column space, Null space, Inverses Celebrity: The Rank Solution concept Some things of Eigen Eigen values, Eigen vectors, Change of Basis Summary 2 Essence of Linear Algebra, Chapter 3

Essence of Linear The Storyline Algebra Shreedhar Kodate The Geometry of Linear Equations The Geometry of Vectors and Basis vectors Linear Equations Vectors and Basis Linear combinations and Span vectors Linear combinations and Span The box game: Matrices The box game: Matrices Elimination and Multiplication, A=LU Elimination and Multiplication, A=LU Transforming your LIFE Leenearly Transforming your LIFE Leenearly Cool Video, The Determinant Cool Video, The Determinant Space Tour Space Tour Column space, Null space, Inverses Column space, Null space, Inverses Celebrity: The Rank Celebrity: The Rank Solution concept Some things of Solution concept Eigen Eigen values, Eigen vectors, Change of Some things of Eigen Basis Eigen values, Eigen vectors, Change of Basis Summary

Essence of Linear Linear combinations Algebra Shreedhar Kodate Additivity The Geometry of Linear Equations       Vectors and Basis  1  6  7 vectors  +  = u + # # » v = # » » w Linear combinations  and Span 2 3 5 The box game: Matrices Elimination and Multiplication, A=LU Scaling Transforming your LIFE Leenearly Cool Video, The     Determinant  1  2 v = # »  = Space Tour 2 # » 2 (2 v )  Column space, Null 2 4 space, Inverses Celebrity: The Rank Solution concept Hybrid Some things of Eigen Eigen values, Eigen vectors, Change of       Basis  1  6  20  + 3  = a # u + b # » v = # » » 2 w Summary  2 3 13

Essence of Linear Span Algebra Shreedhar Kodate The Geometry of Example Linear Equations Vectors and Basis The image 3 below shows how two vectors can span the 2D vectors Linear combinations and Span space. The box game: Matrices Elimination and Multiplication, A=LU Transforming your LIFE Leenearly Cool Video, The Determinant Space Tour Column space, Null space, Inverses Celebrity: The Rank Solution concept Some things of Eigen Eigen values, Eigen vectors, Change of Basis Summary 3 Essence of Linear Algebra, Chapter 2