math 221 linear algebra
play

Math 221: LINEAR ALGEBRA Chapter 3. Determinants and Diagonalization - PowerPoint PPT Presentation

Aug 30, 2023 •9 likes •1.41k views

Math 221: LINEAR ALGEBRA Chapter 3. Determinants and Diagonalization 3-3. Determinants and Diagonalization Le Chen 1 Emory University, 2020 Fall (last updated on 10/21/2020) Creative Commons License (CC BY-NC-SA) 1 Slides are adapted from

  1. Example (continued) Therefore, the eigenvalues of A are λ 1 = 3 , λ 2 = 2 , and λ 3 = 1 . x = � Basic eigenvectors corresponding to λ 1 = 3 : solve (3 I − A ) � 0 . − 1    1 0 0  0 4 − 2 0 2  → · · · → − 1 5 − 2 0 − 1 0 1 0    2 − 1 5 − 2 0 0 0 0 0 1 1  2 t    2  = t 1 1 Thus � x = 2 t  , t ∈ R .   2 t 1  1   as a basic eigenvector corresponding to Choosing t = 2 gives us � x 1 = 1  2 λ 1 = 3 .

  2. Example (continued) x = � Basic eigenvectors corresponding to λ 2 = 2 : solve (2 I − A ) � 0 .    1 0 − 2 0  − 1 4 − 2 0  → · · · → − 1 4 − 2 0 0 1 − 1 0    − 1 5 − 3 0 0 0 0 0

  3. Example (continued) x = � Basic eigenvectors corresponding to λ 2 = 2 : solve (2 I − A ) � 0 .    1 0 − 2 0  − 1 4 − 2 0  → · · · → − 1 4 − 2 0 0 1 − 1 0    − 1 5 − 3 0 0 0 0 0 2 s    2   = s Thus � x =  , s ∈ R . s 1   s 1

  4. Example (continued) x = � Basic eigenvectors corresponding to λ 2 = 2 : solve (2 I − A ) � 0 .    1 0 − 2 0  − 1 4 − 2 0  → · · · → − 1 4 − 2 0 0 1 − 1 0    − 1 5 − 3 0 0 0 0 0 2 s    2   = s Thus � x =  , s ∈ R . s 1   s 1  2   as a basic eigenvector corresponding to Choosing s = 1 gives us � x 2 = 1  1 λ 2 = 2 .

  5. Example (continued) x = � Basic eigenvectors corresponding to λ 3 = 1 : solve ( I − A ) � 0 .    1 0 − 1 0  − 2 4 − 2 0  → · · · → − 1 3 − 2 0 0 1 − 1 0    − 1 5 − 4 0 0 0 0 0

  6. Example (continued) x = � Basic eigenvectors corresponding to λ 3 = 1 : solve ( I − A ) � 0 .    1 0 − 1 0  − 2 4 − 2 0  → · · · → − 1 3 − 2 0 0 1 − 1 0    − 1 5 − 4 0 0 0 0 0 r    1   = r Thus � x =  , r ∈ R . r 1   r 1

  7. Example (continued) x = � Basic eigenvectors corresponding to λ 3 = 1 : solve ( I − A ) � 0 .    1 0 − 1 0  − 2 4 − 2 0  → · · · → − 1 3 − 2 0 0 1 − 1 0    − 1 5 − 4 0 0 0 0 0 r    1   = r Thus � x =  , r ∈ R . r 1   r 1  1   as a basic eigenvector corresponding to Choosing r = 1 gives us � x 3 = 1  1 λ 3 = 1 .

  8. Geometric Interpretation of Eigenvalues and Eigenvectors Let A be a 2 × 2 matrix. Then A can be interpreted as a linear transformation from R 2 to R 2 .

  9. Geometric Interpretation of Eigenvalues and Eigenvectors Let A be a 2 × 2 matrix. Then A can be interpreted as a linear transformation from R 2 to R 2 . Problem How does the linear transformation affect the eigenvectors of the matrix?

  10. Geometric Interpretation of Eigenvalues and Eigenvectors Let A be a 2 × 2 matrix. Then A can be interpreted as a linear transformation from R 2 to R 2 . Problem How does the linear transformation affect the eigenvectors of the matrix? Definition � a � be a nonzero vector in R 2 . Then L � Let � v = v is the set of all scalar b multiples of � v, i.e., L � v = R � v = { t � v | t ∈ R } .

  11. Example (revisited) � 4 − 2 � A = has two eigenvalues: λ 1 = 2 and λ 2 = 5 with − 1 3 corresponding eigenvectors � 1 � � − 1 � v 1 = � and � v 2 = 1 1/2

  12. � 4 − 2 � A = − 1 3 y 6 5 4 3 2 1 x − 5 − 4 − 3 − 2 − 1 1 2 3 4 5 6 − 1 − 2 − 3 − 4

  13. � 4 − 2 � A = − 1 3 y 6 5 4 3 2 1 x − 5 − 4 − 3 − 2 − 1 1 2 3 4 5 6 − 1 − 2 − 3 − 4

  14. � 4 − 2 � A = − 1 3 y 6 5 4 3 2 1 x − 5 − 4 − 3 − 2 − 1 1 2 3 4 5 6 − 1 − 2 − 3 − 4

  15. � 4 − 2 � A = − 1 3 y 6 5 4 3 2 1 x − 5 − 4 − 3 − 2 − 1 1 2 3 4 5 6 − 1 − 2 − 3 − 4

  16. � 4 − 2 � A = − 1 3 y 6 5 4 3 2 1 x − 5 − 4 − 3 − 2 − 1 1 2 3 4 5 6 − 1 − 2 − 3 − 4

  17. � 4 − 2 � A = − 1 3 y 6 5 4 3 2 1 x − 5 − 4 − 3 − 2 − 1 1 2 3 4 5 6 − 1 − 2 − 3 − 4

  18. � 4 − 2 � A = − 1 3 y 6 5 4 3 2 1 x − 5 − 4 − 3 − 2 − 1 1 2 3 4 5 6 − 1 − 2 − 3 − 4

  19. � 4 − 2 � A = − 1 3 y 6 5 4 3 2 1 x − 5 − 4 − 3 − 2 − 1 1 2 3 4 5 6 − 1 − 2 − 3 − 4

  20. � 4 − 2 � A = − 1 3 y 6 5 4 3 2 1 x − 5 − 4 − 3 − 2 − 1 1 2 3 4 5 6 − 1 − 2 − 3 − 4

  21. � 4 − 2 � A = − 1 3 y 6 5 4 3 2 1 x − 5 − 4 − 3 − 2 − 1 1 2 3 4 5 6 − 1 − 2 − 3 − 4

  22. � 4 − 2 � A = − 1 3 y 6 5 4 3 2 1 x − 5 − 4 − 3 − 2 − 1 1 2 3 4 5 6 − 1 − 2 − 3 − 4

  23. � 4 − 2 � A = − 1 3 y 6 5 4 3 2 1 x − 5 − 4 − 3 − 2 − 1 1 2 3 4 5 6 − 1 − 2 − 3 − 4

  24. � 4 − 2 � A = − 1 3 y 6 5 4 3 2 1 x − 5 − 4 − 3 − 2 − 1 1 2 3 4 5 6 − 1 − 2 − 3 − 4

  25. � 4 − 2 � A = − 1 3 y 6 5 4 3 2 1 x − 5 − 4 − 3 − 2 − 1 1 2 3 4 5 6 − 1 − 2 − 3 − 4

  26. Definition Let A be a 2 × 2 matrix and L a line in R 2 through the origin. Then L is said to be A-invariant if the vector A � x lies in L whenever � x lies in L,

  27. Definition Let A be a 2 × 2 matrix and L a line in R 2 through the origin. Then L is said to be A-invariant if the vector A � x lies in L whenever � x lies in L, i.e., A � x is a scalar multiple of � x,

  28. Definition Let A be a 2 × 2 matrix and L a line in R 2 through the origin. Then L is said to be A-invariant if the vector A � x lies in L whenever � x lies in L, i.e., A � x is a scalar multiple of � x, i.e., A � x = λ� x for some scalar λ ∈ R ,

  29. Definition Let A be a 2 × 2 matrix and L a line in R 2 through the origin. Then L is said to be A-invariant if the vector A � x lies in L whenever � x lies in L, i.e., A � x is a scalar multiple of � x, i.e., A � x = λ� x for some scalar λ ∈ R , i.e., � x is an eigenvector of A.

  30. Definition Let A be a 2 × 2 matrix and L a line in R 2 through the origin. Then L is said to be A-invariant if the vector A � x lies in L whenever � x lies in L, i.e., A � x is a scalar multiple of � x, i.e., A � x = λ� x for some scalar λ ∈ R , i.e., � x is an eigenvector of A. Theorem (A-Invariance) v � = 0 be a vector in R 2 . Then L � Let A be a 2 × 2 matrix and let � v is A-invariant if and only if � v is an eigenvector of A.

  31. � 4 − 2 � A = − 1 3 y 6 5 4 3 2 1 x − 5 − 4 − 3 − 2 − 1 1 2 3 4 5 6 − 1 − 2 − 3 − 4

  32. � 4 − 2 � A = − 1 3 y 6 5 4 3 2 1 x − 5 − 4 − 3 − 2 − 1 1 2 3 4 5 6 − 1 − 2 − 3 − 4

  33. � 4 − 2 � A = − 1 3 y 6 5 4 3 2 1 x − 5 − 4 − 3 − 2 − 1 1 2 3 4 5 6 − 1 − 2 − 3 − 4

  34. Problem Let m ∈ R and consider the linear transformation Q m : R 2 → R 2 , i.e., reflection in the line y = mx. y y = mx x Recall that this is a matrix transformation induced by � 1 − m 2 1 � 2 m A = . m 2 − 1 1 + m 2 2 m Find the lines that pass through origin and are A-invariant. Determine corresponding eigenvalues.

  35. Solution y y = mx x

  36. Solution y y = mx x

  37. Solution y y = mx x

  38. Solution y y = mx x

  39. Solution y y = mx x � 1 � Let � x 1 = . Then L � x 1 is A-invariant, that is, � x 1 is an eigenvector. m Since the vector won’t change, its eigenvalue should be 1 . Indeed, one can verify that � 1 − m 2 � � 1 � 1 1 � � 2 m A � x 1 = = ... = = � x 1 . m 2 − 1 1 + m 2 2 m m m

  40. Solution (continued) y y = mx x

  41. Solution (continued) y y = mx x

  42. Solution (continued) y y = mx x

  43. Solution (continued) y y = mx x

  44. Solution (continued) y y = mx x � − m � Let � x 2 = . Then L � x 2 is A-invariant, that is, � x 2 is an eigenvector. 1 Since the vector won’t change the size, only flip the direction, its eigenvalue should be − 1 . Indeed, one can verify that � 1 − m 2 � m 1 � � − m � � 2 m A � x 2 = = · · · = = − � x 2 . m 2 − 1 1 − 1 1 + m 2 2 m

  45. Example Let θ be a real number, and R θ : R 2 → R 2 rotation through an angle of θ , induced by the matrix � cos θ � − sin θ A = . sin θ cos θ

  46. Example Let θ be a real number, and R θ : R 2 → R 2 rotation through an angle of θ , induced by the matrix � cos θ � − sin θ A = . sin θ cos θ Claim: A has no real eigenvalues unless θ is an integer multiple of π , i.e., ± π, ± 2 π, ± 3 π , etc.

  47. Example Let θ be a real number, and R θ : R 2 → R 2 rotation through an angle of θ , induced by the matrix � cos θ � − sin θ A = . sin θ cos θ Claim: A has no real eigenvalues unless θ is an integer multiple of π , i.e., ± π, ± 2 π, ± 3 π , etc. Consequence: a line L in R 2 is A invariant if and only if θ is an integer multiple of π .

  48. Diagonalization Denote an n × n diagonal matrix by  a 1 0 0 · · · 0 0  0 a 2 0 · · · 0 0    0 0 a 3 · · · 0 0    diag ( a 1 , a 2 , . . . , a n ) =  . . . . . .  . . . . . .   . . . . . .     0 0 0 · · · a n − 1 0   0 0 0 · · · 0 a n

  49. Diagonalization Denote an n × n diagonal matrix by  a 1 0 0 · · · 0 0  0 a 2 0 · · · 0 0    0 0 a 3 · · · 0 0    diag ( a 1 , a 2 , . . . , a n ) =  . . . . . .  . . . . . .   . . . . . .     0 0 0 · · · a n − 1 0   0 0 0 · · · 0 a n Recall that if A is an n × n matrix and P is an invertible n × n matrix so that P − 1 AP is diagonal, then P is called a diagonalizing matrix of A, and A is diagonalizable.

  50. columns vectors into a matrix: Pack the above ... ◮ Suppose we have n eigenvalue-eigenvector pairs: A � x j = λ j � x j , j = 1 , 2 , . . . , n

  51. ... ◮ Suppose we have n eigenvalue-eigenvector pairs: A � x j = λ j � x j , j = 1 , 2 , . . . , n ◮ Pack the above n columns vectors into a matrix: � � � � A � x 1 A � x 2 · · · A � x n = λ 1 � x 1 λ 2 � x 2 · · · λ n � x n || � � A � x 1 � x 2 · · · � x n ||  λ 1  λ 2   � x 1 x 2 · · · x n � � � �       λ n

  52. ◮ By denoting: � � P = � x 1 � x 2 · · · � x n and D = diag ( λ 1 , · · · , λ n ) we see that AP = PD

  53. ◮ By denoting: � � P = � x 1 � x 2 · · · � x n and D = diag ( λ 1 , · · · , λ n ) we see that AP = PD ◮ Hence, provided P is invertible, we have A = PDP − 1 D = P − 1 AP or equivalently

  54. ◮ By denoting: � � P = � x 1 � x 2 · · · � x n and D = diag ( λ 1 , · · · , λ n ) we see that AP = PD ◮ Hence, provided P is invertible, we have A = PDP − 1 D = P − 1 AP or equivalently that is, A is diagonalizable.

  55. Theorem (Matrix Diagonalization) Let A be an n × n matrix. 1. A is diagonalizable if and only if it has eigenvectors � x 1 ,� x 2 , . . . ,� x n so that � � P = � x 1 � x 2 · · · � x n is invertible.

  56. Theorem (Matrix Diagonalization) Let A be an n × n matrix. 1. A is diagonalizable if and only if it has eigenvectors � x 1 ,� x 2 , . . . ,� x n so that � � P = � x 1 x 2 � · · · � x n is invertible. 2. If P is invertible, then P − 1 AP = diag ( λ 1 , λ 2 , . . . , λ n ) where λ i is the eigenvalue of A corresponding to the eigenvector � x i , i.e., A � x i = λ i � x i .

  57. Example   3 − 4 2  has eigenvalues and corresponding basic eigenvectors A = 1 − 2 2  1 − 5 5   1  ; λ 1 = 3 and � x 1 = 1  2  2   ; λ 2 = 2 and x 2 = 1 �  1  1   . λ 3 = 1 and � x 3 = 1  1

  58. Example (continued)   1 2 1 � x 1 x 2 x 3 � 1 1 1 Let P = � � � =  .  2 1 1

  59. Example (continued)   1 2 1 � x 1 x 2 x 3 � 1 1 1 Let P = � � � =  .  2 1 1

  60. Example (continued)   1 2 1 � x 1 x 2 x 3 � 1 1 1 Let P = � � � =  . Then P is invertible (check  2 1 1 this!), so by the above Theorem,  3 0 0  P − 1 AP = diag (3 , 2 , 1) =  . 0 2 0  0 0 1

  61. Remark It is not always possible to find n eigenvectors so that P is invertible.

  62. Remark It is not always possible to find n eigenvectors so that P is invertible. Example   1 − 2 3 Let A = 2 6 − 6  .  1 2 − 1

  63. Remark It is not always possible to find n eigenvectors so that P is invertible. Example   1 − 2 3 Let A = 2 6 − 6  .  1 2 − 1 Then � � x − 1 2 − 3 � � � � = · · · = ( x − 2) 3 . − 2 x − 6 6 c A ( x ) = � � � � − 1 − 2 x + 1 � �

  64. Remark It is not always possible to find n eigenvectors so that P is invertible. Example   1 − 2 3 Let A = 2 6 − 6  .  1 2 − 1 Then � � x − 1 2 − 3 � � � � = · · · = ( x − 2) 3 . − 2 x − 6 6 c A ( x ) = � � � � − 1 − 2 x + 1 � � A has only one eigenvalue, λ 1 = 2 , with multiplicity three. Sometimes, one writes λ 1 = λ 2 = λ 3 = 2 .

  65. Example (continued) x = � To find the 2 -eigenvectors of A, solve the system (2 I − A ) � 0 .     1 2 − 3 0 1 2 − 3 0  → · · · → − 2 − 4 6 0 0 0 0 0    − 1 − 2 3 0 0 0 0 0

  66. Example (continued) x = � To find the 2 -eigenvectors of A, solve the system (2 I − A ) � 0 .     1 2 − 3 0 1 2 − 3 0  → · · · → − 2 − 4 6 0 0 0 0 0    − 1 − 2 3 0 0 0 0 0 The general solution in parametric form is       − 2 s + 3 t − 2 3  = s  + t  , s , t ∈ R . � x = s 1 0    t 0 1

  67. Example (continued) x = � To find the 2 -eigenvectors of A, solve the system (2 I − A ) � 0 .     1 2 − 3 0 1 2 − 3 0  → · · · → − 2 − 4 6 0 0 0 0 0    − 1 − 2 3 0 0 0 0 0 The general solution in parametric form is       − 2 s + 3 t − 2 3  = s  + t  , s , t ∈ R . � x = s 1 0    t 0 1 Since the system has only two basic solutions, there are only two basic eigenvectors, implying that the matrix A is not diagonalizable.

  68. det Example   1 0 1 Diagonalize, if possible, the matrix A = 0 1 0  .  0 0 − 3

  69. Example   1 0 1 Diagonalize, if possible, the matrix A = 0 1 0  .  0 0 − 3 � � x − 1 0 − 1 � � � � = ( x − 1) 2 ( x + 3) . c A ( x ) = det ( xI − A ) = 0 x − 1 0 � � � � 0 0 x + 3 � �

Recommend

Math you need to know 1 Linear algebra Linear algebra is mainly concerned with solving equations

Math you need to know 1 Linear algebra Linear algebra is mainly concerned with solving equations

Peter Latham, September 30, 2014 1 Math you need to know 1 Linear algebra Linear algebra is mainly concerned with solving equations of the form A x = y , (1) { lin_eq } which is written in terms of components as A ij x j = y j . (2)

273 views • 6 slides
Math you need to know 1 Linear algebra Linear algebra is mainly concerned with solving equations

Math you need to know 1 Linear algebra Linear algebra is mainly concerned with solving equations

Peter Latham, October 7, 2014 1 Math you need to know 1 Linear algebra Linear algebra is mainly concerned with solving equations of the form A x = y , (1) { lin_eq } which is written in terms of components as A ij x j = y j . (2) j

477 views • 11 slides
Linear Algebra II: linear combinations & matrices Math Tools for Neuroscience (NEU 314)

Linear Algebra II: linear combinations & matrices Math Tools for Neuroscience (NEU 314)

Linear Algebra II: linear combinations & matrices Math Tools for Neuroscience (NEU 314) Fall 2016 Jonathan Pillow Princeton Neuroscience Institute & Psychology. Lecture 3 (Thursday 9/22) accompanying notes/slides Linear

255 views • 10 slides
Math 221: LINEAR ALGEBRA Chapter 7. Linear Transformations 7-3. Isomorphisms and Composition Le

Math 221: LINEAR ALGEBRA Chapter 7. Linear Transformations 7-3. Isomorphisms and Composition Le

Math 221: LINEAR ALGEBRA Chapter 7. Linear Transformations 7-3. Isomorphisms and Composition Le Chen 1 Emory University, 2020 Fall (last updated on 10/21/2020) Creative Commons License (CC BY-NC-SA) 1 Slides are adapted from those by Karen

861 views • 59 slides
Linear Algebra Linear algebra has become as basic and as applicable as calculus, and

Linear Algebra Linear algebra has become as basic and as applicable as calculus, and

Mathematical Tools for Neural and Cognitive Science Fall semester, 2018 Section 1: Linear Algebra Linear Algebra Linear algebra has become as basic and as applicable as calculus, and fortunately it is easier - Gilbert Strang, Linear

359 views • 10 slides
Math 221: LINEAR ALGEBRA Chapter 1. Systems of Linear Equations 1-1. Solutions and Elementary

Math 221: LINEAR ALGEBRA Chapter 1. Systems of Linear Equations 1-1. Solutions and Elementary

Math 221: LINEAR ALGEBRA Chapter 1. Systems of Linear Equations 1-1. Solutions and Elementary Operations Le Chen 1 Emory University, 2020 Fall (last updated on 10/09/2020) Creative Commons License (CC BY-NC-SA) 1 Slides are adapted from

936 views • 70 slides
Math 221: LINEAR ALGEBRA 7-2. Linear Transformations - Kernel and Image Le Chen 1 Emory

Math 221: LINEAR ALGEBRA 7-2. Linear Transformations - Kernel and Image Le Chen 1 Emory

Math 221: LINEAR ALGEBRA 7-2. Linear Transformations - Kernel and Image Le Chen 1 Emory University, 2020 Fall (last updated on 08/27/2020) Creative Commons License (CC BY-NC-SA) 1 Slides are adapted from those by Karen Seyffarth from

861 views • 64 slides
Math 221: LINEAR ALGEBRA Chapter 1. Systems of Linear Equations 1-3. Homogeneous Equations Le

Math 221: LINEAR ALGEBRA Chapter 1. Systems of Linear Equations 1-3. Homogeneous Equations Le

Math 221: LINEAR ALGEBRA Chapter 1. Systems of Linear Equations 1-3. Homogeneous Equations Le Chen 1 Emory University, 2020 Fall (last updated on 10/21/2020) Creative Commons License (CC BY-NC-SA) 1 Slides are adapted from those by Karen

415 views • 25 slides
Math 221: LINEAR ALGEBRA Chapter 2. Matrix Algebra 2-9. Markov Chains Le Chen 1 Emory

Math 221: LINEAR ALGEBRA Chapter 2. Matrix Algebra 2-9. Markov Chains Le Chen 1 Emory

Math 221: LINEAR ALGEBRA Chapter 2. Matrix Algebra 2-9. Markov Chains Le Chen 1 Emory University, 2020 Fall (last updated on 10/09/2020) Creative Commons License (CC BY-NC-SA) 1 Slides are adapted from those by Karen Seyffarth from

722 views • 53 slides
Math 221: LINEAR ALGEBRA Chapter 2. Matrix Algebra 2-1. Matrix Addition, Scalar Multiplication

Math 221: LINEAR ALGEBRA Chapter 2. Matrix Algebra 2-1. Matrix Addition, Scalar Multiplication

Math 221: LINEAR ALGEBRA Chapter 2. Matrix Algebra 2-1. Matrix Addition, Scalar Multiplication and Transposition Le Chen 1 Emory University, 2020 Fall (last updated on 10/21/2020) Creative Commons License (CC BY-NC-SA) 1 Slides are adapted

998 views • 57 slides
Math 313 - Linear Algebra 1.7 - 1.10 September 9 - 16, 2016 There is no royal road to

Math 313 - Linear Algebra 1.7 - 1.10 September 9 - 16, 2016 There is no royal road to

Math 313 - Linear Algebra 1.7 - 1.10 September 9 - 16, 2016 There is no royal road to geometry. - Euclid to Ptolemy 1.7 Linear Independence Definition (Linear Independence) An indexed set of vectors { v 1 , . . . , v p } in R n is said to

393 views • 26 slides
Math 221: LINEAR ALGEBRA 3-4. Application to Linear Recurrences Le Chen 1 Emory University,

Math 221: LINEAR ALGEBRA 3-4. Application to Linear Recurrences Le Chen 1 Emory University,

Math 221: LINEAR ALGEBRA 3-4. Application to Linear Recurrences Le Chen 1 Emory University, 2020 Fall (last updated on 08/27/2020) Creative Commons License (CC BY-NC-SA) 1 Slides are adapted from those by Karen Seyffarth from University of

444 views • 25 slides
Deep Learning Prof. Kuan-Ting Lai 2020/3/10 Applied Math for Deep Learning Linear Algebra

Deep Learning Prof. Kuan-Ting Lai 2020/3/10 Applied Math for Deep Learning Linear Algebra

Applied Math for Deep Learning Prof. Kuan-Ting Lai 2020/3/10 Applied Math for Deep Learning Linear Algebra Probability Calculus Optimization Linear Algebra Scalar real numbers Vector (1D) Has a magnitude & a

1.19k views • 69 slides
Math 221: LINEAR ALGEBRA Chapter 6. Vector Spaces 6-3. Vector Spaces - Linear Independence Le

Math 221: LINEAR ALGEBRA Chapter 6. Vector Spaces 6-3. Vector Spaces - Linear Independence Le

Math 221: LINEAR ALGEBRA Chapter 6. Vector Spaces 6-3. Vector Spaces - Linear Independence Le Chen 1 Emory University, 2020 Fall (last updated on 10/09/2020) Creative Commons License (CC BY-NC-SA) 1 Slides are adapted from those by Karen

666 views • 50 slides
Basic Concepts in Linear Algebra Department of Mathematics Boise State University September 14,

Basic Concepts in Linear Algebra Department of Mathematics Boise State University September 14,

Basic Concepts in Linear Algebra Department of Mathematics Boise State University September 14, 2017 Math 365 Linear Algebra Basics September 14, 2017 1 / 39 Numerical Linear Algebra Linear systems of equations occur in almost every area of

667 views • 47 slides
Chapter 1 What is Linear Algebra? Chapter 1 What is Linear Algebra? The study of linear

Chapter 1 What is Linear Algebra? Chapter 1 What is Linear Algebra? The study of linear

Chapter 1 What is Linear Algebra? Chapter 1 What is Linear Algebra? The study of linear functions. The word linear means straight or flat . y = 0 + 1 x Linear functions involve only addition and scalar multiplication. Chapter 1 Higher

1.2k views • 56 slides
Linear Algebra III: vector spaces Math Tools for Neuroscience (NEU 314) Fall 2016 Jonathan

Linear Algebra III: vector spaces Math Tools for Neuroscience (NEU 314) Fall 2016 Jonathan

Linear Algebra III: vector spaces Math Tools for Neuroscience (NEU 314) Fall 2016 Jonathan Pillow Princeton Neuroscience Institute & Psychology. Lecture 4 (Tuesday 9/27) accompanying notes/slides Outline Last time: linear

494 views • 17 slides
Machine Learning Prof. Kuan-Ting Lai 2020/4/11 Applied Math for Machine Learning Linear

Machine Learning Prof. Kuan-Ting Lai 2020/4/11 Applied Math for Machine Learning Linear

Applied Math for Machine Learning Prof. Kuan-Ting Lai 2020/4/11 Applied Math for Machine Learning Linear Algebra Probability Calculus Optimization Linear Algebra Scalar real numbers Vector (1D) Has a magnitude

736 views • 54 slides
Math 221: LINEAR ALGEBRA 2-2. Equations, Matrices, and Transformations Le Chen 1 Emory

Math 221: LINEAR ALGEBRA 2-2. Equations, Matrices, and Transformations Le Chen 1 Emory

Math 221: LINEAR ALGEBRA 2-2. Equations, Matrices, and Transformations Le Chen 1 Emory University, 2020 Fall (last updated on 09/02/2020) Creative Commons License (CC BY-NC-SA) 1 Slides are adapted from those by Karen Seyffarth from

1k views • 73 slides
Linear algebra explained in four pages Excerpt from the N O BULLSHIT GUIDE TO LINEAR ALGEBRA by

Linear algebra explained in four pages Excerpt from the N O BULLSHIT GUIDE TO LINEAR ALGEBRA by

Linear algebra explained in four pages Excerpt from the N O BULLSHIT GUIDE TO LINEAR ALGEBRA by Ivan Savov Abstract This document will review the fundamental ideas of linear algebra. B. Matrix operations We will learn about matrices, matrix

333 views • 4 slides
Linear Algebra for Machine Learning Sargur N. Srihari srihari@cedar.buffalo.edu 1 Deep

Linear Algebra for Machine Learning Sargur N. Srihari srihari@cedar.buffalo.edu 1 Deep

Deep Learning Srihari Linear Algebra for Machine Learning Sargur N. Srihari srihari@cedar.buffalo.edu 1 Deep Learning Srihari Overview Linear Algebra is based on continuous math rather than discrete math Computer scientists have

841 views • 50 slides
Linear algebra and differential equations (Math 54): Lecture 5 Vivek Shende February 5, 2019

Linear algebra and differential equations (Math 54): Lecture 5 Vivek Shende February 5, 2019

Linear algebra and differential equations (Math 54): Lecture 5 Vivek Shende February 5, 2019 Hello and welcome to class! Hello and welcome to class! Last time We discussed linear transformations, and their matrix representations. Hello and

2.07k views • 137 slides
Linear algebra fundamentals Matthew Macauley Department of Mathematical Sciences Clemson

Linear algebra fundamentals Matthew Macauley Department of Mathematical Sciences Clemson

Linear algebra fundamentals Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 8530, Spring 2017 M. Macauley (Clemson) Linear algebra fundamentals Math 8530, Spring 2017 1 / 17

528 views • 17 slides
Math 221: LINEAR ALGEBRA 5-1. Vector Space R n - Subspaces and Spanning Le Chen 1 Emory

Math 221: LINEAR ALGEBRA 5-1. Vector Space R n - Subspaces and Spanning Le Chen 1 Emory

Math 221: LINEAR ALGEBRA 5-1. Vector Space R n - Subspaces and Spanning Le Chen 1 Emory University, 2020 Fall (last updated on 08/18/2020) Creative Commons License (CC BY-NC-SA) 1 Slides are adapted from those by Karen Seyffarth from

997 views • 61 slides

More recommend