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Raffi Hovasapian
Orthogonal Complements, Part II
Slide Duration:Table of Contents
I. Linear Equations and Matrices
Linear Systems
39m 3s
 Intro0:00
 Linear Systems1:20
 Introduction to Linear Systems1:21
 Examples10:35
 Example 110:36
 Example 213:44
 Example 316:12
 Example 423:48
 Example 528:23
 Example 632:32
 Number of Solutions35:08
 One Solution, No Solution, Infinitely Many Solutions35:09
 Method of Elimination36:57
 Method of Elimination36:58
Matrices
30m 34s
 Intro0:00
 Matrices0:47
 Definition and Example of Matrices0:48
 Square Matrix7:55
 Diagonal Matrix9:31
 Operations with Matrices10:35
 Matrix Addition10:36
 Scalar Multiplication15:01
 Transpose of a Matrix17:51
 Matrix Types23:17
 Regular: m x n Matrix of m Rows and n Column23:18
 Square: n x n Matrix With an Equal Number of Rows and Columns23:44
 Diagonal: A Square Matrix Where All Entries OFF the Main Diagonal are '0'24:07
 Matrix Operations24:37
 Matrix Operations24:38
 Example25:55
 Example25:56
Dot Product & Matrix Multiplication
41m 42s
 Intro0:00
 Dot Product1:04
 Example of Dot Product1:05
 Matrix Multiplication7:05
 Definition7:06
 Example 112:26
 Example 217:38
 Matrices and Linear Systems21:24
 Matrices and Linear Systems21:25
 Example 129:56
 Example 232:30
 Summary33:56
 Dot Product of Two Vectors and Matrix Multiplication33:57
 Summary, cont.35:06
 Matrix Representations of Linear Systems35:07
 Examples35:34
 Examples35:35
Properties of Matrix Operation
43m 17s
 Intro0:00
 Properties of Addition1:11
 Properties of Addition: A1:12
 Properties of Addition: B2:30
 Properties of Addition: C2:57
 Properties of Addition: D4:20
 Properties of Addition5:22
 Properties of Addition5:23
 Properties of Multiplication6:47
 Properties of Multiplication: A7:46
 Properties of Multiplication: B8:13
 Properties of Multiplication: C9:18
 Example: Properties of Multiplication9:35
 Definitions and Properties (Multiplication)14:02
 Identity Matrix: n x n matrix14:03
 Let A Be a Matrix of m x n15:23
 Definitions and Properties (Multiplication)18:36
 Definitions and Properties (Multiplication)18:37
 Properties of Scalar Multiplication22:54
 Properties of Scalar Multiplication: A23:39
 Properties of Scalar Multiplication: B24:04
 Properties of Scalar Multiplication: C24:29
 Properties of Scalar Multiplication: D24:48
 Properties of the Transpose25:30
 Properties of the Transpose25:31
 Properties of the Transpose30:28
 Example30:29
 Properties of Matrix Addition33:25
 Let A, B, C, and D Be m x n Matrices33:26
 There is a Unique m x n Matrix, 0, Such That…33:48
 Unique Matrix D34:17
 Properties of Matrix Multiplication34:58
 Let A, B, and C Be Matrices of the Appropriate Size34:59
 Let A Be Square Matrix (n x n)35:44
 Properties of Scalar Multiplication36:35
 Let r and s Be Real Numbers, and A and B Matrices36:36
 Properties of the Transpose37:10
 Let r Be a Scalar, and A and B Matrices37:12
 Example37:58
 Example37:59
Solutions of Linear Systems, Part 1
38m 14s
 Intro0:00
 Reduced Row Echelon Form0:29
 An m x n Matrix is in Reduced Row Echelon Form If:0:30
 Reduced Row Echelon Form2:58
 Example: Reduced Row Echelon Form2:59
 Theorem8:30
 Every m x n Matrix is RowEquivalent to a UNIQUE Matrix in Reduced Row Echelon Form8:31
 Systematic and Careful Example10:02
 Step 110:54
 Step 211:33
 Step 312:50
 Step 414:02
 Step 515:31
 Step 617:28
 Example30:39
 Find the Reduced Row Echelon Form of a Given m x n Matrix30:40
Solutions of Linear Systems, Part II
28m 54s
 Intro0:00
 Solutions of Linear Systems0:11
 Solutions of Linear Systems0:13
 Example I3:25
 Solve the Linear System 13:26
 Solve the Linear System 214:31
 Example II17:41
 Solve the Linear System 317:42
 Solve the Linear System 420:17
 Homogeneous Systems21:54
 Homogeneous Systems Overview21:55
 Theorem and Example24:01
Inverse of a Matrix
40m 10s
 Intro0:00
 Finding the Inverse of a Matrix0:41
 Finding the Inverse of a Matrix0:42
 Properties of NonSingular Matrices6:38
 Practical Procedure9:15
 Step19:16
 Step 210:10
 Step 310:46
 Example: Finding Inverse12:50
 Linear Systems and Inverses17:01
 Linear Systems and Inverses17:02
 Theorem and Example21:15
 Theorem26:32
 Theorem26:33
 List of NonSingular Equivalences28:37
 Example: Does the Following System Have a Nontrivial Solution?30:13
 Example: Inverse of a Matrix36:16
II. Determinants
Determinants
21m 25s
 Intro0:00
 Determinants0:37
 Introduction to Determinants0:38
 Example6:12
 Properties9:00
 Properties 159:01
 Example10:14
 Properties, cont.12:28
 Properties 6 & 712:29
 Example14:14
 Properties, cont.18:34
 Properties 8 & 918:35
 Example19:21
Cofactor Expansions
59m 31s
 Intro0:00
 Cofactor Expansions and Their Application0:42
 Cofactor Expansions and Their Application0:43
 Example 13:52
 Example 27:08
 Evaluation of Determinants by Cofactor9:38
 Theorem9:40
 Example 111:41
 Inverse of a Matrix by Cofactor22:42
 Inverse of a Matrix by Cofactor and Example22:43
 More Example36:22
 List of NonSingular Equivalences43:07
 List of NonSingular Equivalences43:08
 Example44:38
 Cramer's Rule52:22
 Introduction to Cramer's Rule and Example52:23
III. Vectors in Rn
Vectors in the Plane
46m 54s
 Intro0:00
 Vectors in the Plane0:38
 Vectors in the Plane0:39
 Example 18:25
 Example 215:23
 Vector Addition and Scalar Multiplication19:33
 Vector Addition19:34
 Scalar Multiplication24:08
 Example26:25
 The Angle Between Two Vectors29:33
 The Angle Between Two Vectors29:34
 Example33:54
 Properties of the Dot Product and Unit Vectors38:17
 Properties of the Dot Product and Unit Vectors38:18
 Defining Unit Vectors40:01
 2 Very Important Unit Vectors41:56
nVector
52m 44s
 Intro0:00
 nVectors0:58
 4Vector0:59
 7Vector1:50
 Vector Addition2:43
 Scalar Multiplication3:37
 Theorem: Part 14:24
 Theorem: Part 211:38
 Right and Left Handed Coordinate System14:19
 Projection of a Point Onto a Coordinate Line/Plane17:20
 Example21:27
 CauchySchwarz Inequality24:56
 Triangle Inequality36:29
 Unit Vector40:34
 Vectors and Dot Products44:23
 Orthogonal Vectors44:24
 CauchySchwarz Inequality45:04
 Triangle Inequality45:21
 Example 145:40
 Example 248:16
Linear Transformation
48m 53s
 Intro0:00
 Introduction to Linear Transformations0:44
 Introduction to Linear Transformations0:45
 Example 19:01
 Example 211:33
 Definition of Linear Mapping14:13
 Example 322:31
 Example 426:07
 Example 530:36
 Examples36:12
 Projection Mapping36:13
 Images, Range, and Linear Mapping38:33
 Example of Linear Transformation42:02
Linear Transformations, Part II
34m 8s
 Intro0:00
 Linear Transformations1:29
 Linear Transformations1:30
 Theorem 17:15
 Theorem 29:20
 Example 1: Find L (3, 4, 2)11:17
 Example 2: Is It Linear?17:11
 Theorem 325:57
 Example 3: Finding the Standard Matrix29:09
Lines and Planes
37m 54s
 Intro0:00
 Lines and Plane0:36
 Example 10:37
 Example 27:07
 Lines in IR39:53
 Parametric Equations14:58
 Example 317:26
 Example 420:11
 Planes in IR325:19
 Example 531:12
 Example 634:18
IV. Real Vector Spaces
Vector Spaces
42m 19s
 Intro0:00
 Vector Spaces3:43
 Definition of Vector Spaces3:44
 Vector Spaces 15:19
 Vector Spaces 29:34
 Real Vector Space and Complex Vector Space14:01
 Example 115:59
 Example 218:42
 Examples26:22
 More Examples26:23
 Properties of Vector Spaces32:53
 Properties of Vector Spaces Overview32:54
 Property A34:31
 Property B36:09
 Property C36:38
 Property D37:54
 Property F39:00
Subspaces
43m 37s
 Intro0:00
 Subspaces0:47
 Defining Subspaces0:48
 Example 13:08
 Example 23:49
 Theorem7:26
 Example 39:11
 Example 412:30
 Example 516:05
 Linear Combinations23:27
 Definition 123:28
 Example 125:24
 Definition 229:49
 Example 231:34
 Theorem32:42
 Example 334:00
Spanning Set for a Vector Space
33m 15s
 Intro0:00
 A Spanning Set for a Vector Space1:10
 A Spanning Set for a Vector Space1:11
 Procedure to Check if a Set of Vectors Spans a Vector Space3:38
 Example 16:50
 Example 214:28
 Example 321:06
 Example 422:15
Linear Independence
17m 20s
 Intro0:00
 Linear Independence0:32
 Definition0:39
 Meaning3:00
 Procedure for Determining if a Given List of Vectors is Linear Independence or Linear Dependence5:00
 Example 17:21
 Example 210:20
Basis & Dimension
31m 20s
 Intro0:00
 Basis and Dimension0:23
 Definition0:24
 Example 13:30
 Example 2: Part A4:00
 Example 2: Part B6:53
 Theorem 19:40
 Theorem 211:32
 Procedure for Finding a Subset of S that is a Basis for Span S14:20
 Example 316:38
 Theorem 321:08
 Example 425:27
Homogeneous Systems
24m 45s
 Intro0:00
 Homogeneous Systems0:51
 Homogeneous Systems0:52
 Procedure for Finding a Basis for the Null Space of Ax = 02:56
 Example 17:39
 Example 218:03
 Relationship Between Homogeneous and NonHomogeneous Systems19:47
Rank of a Matrix, Part I
35m 3s
 Intro0:00
 Rank of a Matrix1:47
 Definition1:48
 Theorem 18:14
 Example 19:38
 Defining Row and Column Rank16:53
 If We Want a Basis for Span S Consisting of Vectors From S22:00
 If We want a Basis for Span S Consisting of Vectors Not Necessarily in S24:07
 Example 2: Part A26:44
 Example 2: Part B32:10
Rank of a Matrix, Part II
29m 26s
 Intro0:00
 Rank of a Matrix0:17
 Example 1: Part A0:18
 Example 1: Part B5:58
 Rank of a Matrix Review: Rows, Columns, and Row Rank8:22
 Procedure for Computing the Rank of a Matrix14:36
 Theorem 1: Rank + Nullity = n16:19
 Example 217:48
 Rank & Singularity20:09
 Example 321:08
 Theorem 223:25
 List of NonSingular Equivalences24:24
 List of NonSingular Equivalences24:25
Coordinates of a Vector
27m 3s
 Intro0:00
 Coordinates of a Vector1:07
 Coordinates of a Vector1:08
 Example 18:35
 Example 215:28
 Example 3: Part A19:15
 Example 3: Part B22:26
Change of Basis & Transition Matrices
33m 47s
 Intro0:00
 Change of Basis & Transition Matrices0:56
 Change of Basis & Transition Matrices0:57
 Example 110:44
 Example 220:44
 Theorem23:37
 Example 3: Part A26:21
 Example 3: Part B32:05
Orthonormal Bases in nSpace
32m 53s
 Intro0:00
 Orthonormal Bases in nSpace1:02
 Orthonormal Bases in nSpace: Definition1:03
 Example 14:31
 Theorem 16:55
 Theorem 28:00
 Theorem 39:04
 Example 210:07
 Theorem 213:54
 Procedure for Constructing an O/N Basis16:11
 Example 321:42
Orthogonal Complements, Part I
21m 27s
 Intro0:00
 Orthogonal Complements0:19
 Definition0:20
 Theorem 15:36
 Example 16:58
 Theorem 213:26
 Theorem 315:06
 Example 218:20
Orthogonal Complements, Part II
33m 49s
 Intro0:00
 Relations Among the Four Fundamental Vector Spaces Associated with a Matrix A2:16
 Four Spaces Associated With A (If A is m x n)2:17
 Theorem4:49
 Example 17:17
 Null Space and Column Space10:48
 Projections and Applications16:50
 Projections and Applications16:51
 Projection Illustration21:00
 Example 123:51
 Projection Illustration Review30:15
V. Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors
38m 11s
 Intro0:00
 Eigenvalues and Eigenvectors0:38
 Eigenvalues and Eigenvectors0:39
 Definition 13:30
 Example 17:20
 Example 210:19
 Definition 221:15
 Example 323:41
 Theorem 126:32
 Theorem 227:56
 Example 429:14
 Review34:32
Similar Matrices & Diagonalization
29m 55s
 Intro0:00
 Similar Matrices and Diagonalization0:25
 Definition 10:26
 Example 12:00
 Properties3:38
 Definition 24:57
 Theorem 16:12
 Example 39:37
 Theorem 212:40
 Example 419:12
 Example 520:55
 Procedure for Diagonalizing Matrix A: Step 124:21
 Procedure for Diagonalizing Matrix A: Step 225:04
 Procedure for Diagonalizing Matrix A: Step 325:38
 Procedure for Diagonalizing Matrix A: Step 427:02
Diagonalization of Symmetric Matrices
30m 14s
 Intro0:00
 Diagonalization of Symmetric Matrices1:15
 Diagonalization of Symmetric Matrices1:16
 Theorem 12:24
 Theorem 23:27
 Example 14:47
 Definition 16:44
 Example 28:15
 Theorem 310:28
 Theorem 412:31
 Example 318:00
VI. Linear Transformations
Linear Mappings Revisited
24m 5s
 Intro0:00
 Linear Mappings2:08
 Definition2:09
 Linear Operator7:36
 Projection8:48
 Dilation9:40
 Contraction10:07
 Reflection10:26
 Rotation11:06
 Example 113:00
 Theorem 118:16
 Theorem 219:20
Kernel and Range of a Linear Map, Part I
26m 38s
 Intro0:00
 Kernel and Range of a Linear Map0:28
 Definition 10:29
 Example 14:36
 Example 28:12
 Definition 210:34
 Example 313:34
 Theorem 116:01
 Theorem 218:26
 Definition 321:11
 Theorem 324:28
Kernel and Range of a Linear Map, Part II
25m 54s
 Intro0:00
 Kernel and Range of a Linear Map1:39
 Theorem 11:40
 Example 1: Part A2:32
 Example 1: Part B8:12
 Example 1: Part C13:11
 Example 1: Part D14:55
 Theorem 216:50
 Theorem 323:00
Matrix of a Linear Map
33m 21s
 Intro0:00
 Matrix of a Linear Map0:11
 Theorem 11:24
 Procedure for Computing to Matrix: Step 17:10
 Procedure for Computing to Matrix: Step 28:58
 Procedure for Computing to Matrix: Step 39:50
 Matrix of a Linear Map: Property10:41
 Example 114:07
 Example 218:12
 Example 324:31
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For more information, please see full course syllabus of Linear Algebra
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Post by Manfred Berger on June 21, 2013
I've been thinking a bit about example 1. If I was to use GramSchmidt to expand this into a full basis of R3 and then take the image of v with respect to my new basis, the projection would be the first 2 components, correct?