Cofactor Expansions, Linear Algebra

Cofactor Expansions

II. Determinants: Lecture 2 | 59:31 min

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Cofactor Expansions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

Intro

Cofactor Expansions and Their Application

Evaluation of Determinants by Cofactor

Inverse of a Matrix by Cofactor

List of Non-Singular Equivalences

Cramer's Rule

Introduction to Cramer's Rule and Example

I. Linear Equations and Matrices
	Linear Systems	39:03
	Matrices	30:34
	Dot Product & Matrix Multiplication	41:42
	Properties of Matrix Operation	43:17
	Solutions of Linear Systems, Part 1	38:14
	Solutions of Linear Systems, Part II	28:54
	Inverse of a Matrix	28:54
II. Determinants
	Determinants	21:25
	Cofactor Expansions	59:31
III. Vectors in Rn
	Vectors in the Plane	46:54
	n-Vector	52:44
	Linear Transformation	48:53
	Linear Transformations, Part II	34:08
	Lines and Planes	37:54
IV. Real Vector Spaces
	Vector Spaces	42:19
	Subspaces	43:37
	Spanning Set for a Vector Space	33:15
	Linear Independence	17:20
	Basis & Dimension	31:20
	Homogeneous Systems	24:45
	Rank of a Matrix, Part I	35:03
	Rank of a Matrix, Part II	29:26
	Coordinates of a Vector	27:03
	Change of Basis & Transition Matrices	33:47
	Orthonormal Bases in n-Space	32:53
	Orthogonal Complements, Part I	21:27
	Orthogonal Complements, Part II	33:49
V. Eigenvalues and Eigenvectors
	Eigenvalues and Eigenvectors	38:11
	Similar Matrices & Diagonalization	29:55
	Diagonalization of Symmetric Matrices	30:14
VI. Linear Transformations
	Linear Mappings Revisited	24:05
	Kernel and Range of a Linear Map, Part I	24:05
	Kernel and Range of a Linear Map, Part II	25:54
	Matrix of a Linear Map	33:21

Mathematics: Linear Algebra

I. Linear Equations and Matrices
	Linear Systems			39:03
		Intro		0:00
		Linear Systems		1:20
			Introduction to Linear Systems	1:21
		Examples		10:35
			Example 1	10:36
			Example 2	13:44
			Example 3	16:12
			Example 4	23:48
			Example 5	28:23
			Example 6	32:32
		Number of Solutions		35:08
			One Solution, No Solution, Infinitely Many Solutions	35:09
		Method of Elimination		36:57
			Method of Elimination	36:58
	Matrices			30:34
		Intro		0:00
		Matrices		0:47
			Definition and Example of Matrices	0:48
			Square Matrix	7:55
			Diagonal Matrix	9:31
		Operations with Matrices		10:35
			Matrix Addition	10:36
			Scalar Multiplication	15:01
			Transpose of a Matrix	17:51
		Matrix Types		23:17
			Regular: m x n Matrix of m Rows and n Column	23:18
			Square: n x n Matrix With an Equal Number of Rows and Columns	23:44
			Diagonal: A Square Matrix Where All Entries OFF the Main Diagonal are '0'	24:07
		Matrix Operations		24:37
			Matrix Operations	24:38
		Example		25:55
			Example	25:56
	Dot Product & Matrix Multiplication			41:42
		Intro		0:00
		Dot Product		1:04
			Example of Dot Product	1:05
		Matrix Multiplication		7:05
			Definition	7:06
			Example 1	12:26
			Example 2	17:38
		Matrices and Linear Systems		21:24
			Matrices and Linear Systems	21:25
			Example 1	29:56
			Example 2	32:30
		Summary		33:56
			Dot Product of Two Vectors and Matrix Multiplication	33:57
		Summary, cont.		35:06
			Matrix Representations of Linear Systems	35:07
		Examples		35:34
			Examples	35:35
	Properties of Matrix Operation			43:17
		Intro		0:00
		Properties of Addition		1:11
			Properties of Addition: A	1:12
			Properties of Addition: B	2:30
			Properties of Addition: C	2:57
			Properties of Addition: D	4:20
		Properties of Addition		5:22
			Properties of Addition	5:23
		Properties of Multiplication		6:47
			Properties of Multiplication: A	7:46
			Properties of Multiplication: B	8:13
			Properties of Multiplication: C	9:18
			Example: Properties of Multiplication	9:35
		Definitions and Properties (Multiplication)		14:02
			Identity Matrix: n x n matrix	14:03
			Let A Be a Matrix of m x n	15:23
		Definitions and Properties (Multiplication)		18:36
			Definitions and Properties (Multiplication)	18:37
		Properties of Scalar Multiplication		22:54
			Properties of Scalar Multiplication: A	23:39
			Properties of Scalar Multiplication: B	24:04
			Properties of Scalar Multiplication: C	24:29
			Properties of Scalar Multiplication: D	24:48
		Properties of the Transpose		25:30
			Properties of the Transpose	25:31
		Properties of the Transpose		30:28
			Example	30:29
		Properties of Matrix Addition		33:25
			Let A, B, C, and D Be m x n Matrices	33:26
			There is a Unique m x n Matrix, 0, Such That…	33:48
			Unique Matrix D	34:17
		Properties of Matrix Multiplication		34:58
			Let A, B, and C Be Matrices of the Appropriate Size	34:59
			Let A Be Square Matrix (n x n)	35:44
		Properties of Scalar Multiplication		36:35
			Let r and s Be Real Numbers, and A and B Matrices	36:36
		Properties of the Transpose		37:10
			Let r Be a Scalar, and A and B Matrices	37:12
		Example		37:58
			Example	37:59
	Solutions of Linear Systems, Part 1			38:14
		Intro		0:00
		Reduced Row Echelon Form		0:29
			An m x n Matrix is in Reduced Row Echelon Form If:	0:30
		Reduced Row Echelon Form		2:58
			Example: Reduced Row Echelon Form	2:59
		Theorem		8:30
			Every m x n Matrix is Row-Equivalent to a UNIQUE Matrix in Reduced Row Echelon Form	8:31
			Systematic and Careful Example	10:02
			Step 1	10:54
			Step 2	11:33
			Step 3	12:50
			Step 4	14:02
			Step 5	15:31
			Step 6	17:28
		Example		30:39
			Find the Reduced Row Echelon Form of a Given m x n Matrix	30:40
	Solutions of Linear Systems, Part II			28:54
		Intro		0:00
		Solutions of Linear Systems		0:11
			Solutions of Linear Systems	0:13
		Example I		3:25
			Solve the Linear System 1	3:26
			Solve the Linear System 2	14:31
		Example II		17:41
			Solve the Linear System 3	17:42
			Solve the Linear System 4	20:17
		Homogeneous Systems		21:54
			Homogeneous Systems Overview	21:55
			Theorem and Example	24:01
	Inverse of a Matrix			28:54
		Intro		0:00
		Finding the Inverse of a Matrix		0:41
			Finding the Inverse of a Matrix	0:42
			Properties of Non-Singular Matrices	6:38
		Practical Procedure		9:15
			Step1	9:16
			Step 2	10:10
			Step 3	10:46
			Example: Finding Inverse	12:50
		Linear Systems and Inverses		17:01
			Linear Systems and Inverses	17:02
			Theorem and Example	21:15
		Theorem		26:32
			Theorem	26:33
			List of Non-Singular Equivalences	28:37
			Example: Does the Following System Have a Non-trivial Solution?	30:13
			Example: Inverse of a Matrix	36:16
II. Determinants
	Determinants			21:25
		Intro		0:00
		Determinants		0:37
			Introduction to Determinants	0:38
			Example	6:12
		Properties		9:00
			Properties 1-5	9:01
			Example	10:14
		Properties, cont.		12:28
			Properties 6 & 7	12:29
			Example	14:14
		Properties, cont.		18:34
			Properties 8 & 9	18:35
			Example	19:21
	Cofactor Expansions			59:31
		Intro		0:00
		Cofactor Expansions and Their Application		0:42
			Cofactor Expansions and Their Application	0:43
			Example 1	3:52
			Example 2	7:08
		Evaluation of Determinants by Cofactor		9:38
			Theorem	9:40
			Example 1	11:41
		Inverse of a Matrix by Cofactor		22:42
			Inverse of a Matrix by Cofactor and Example	22:43
			More Example	36:22
		List of Non-Singular Equivalences		43:07
			List of Non-Singular Equivalences	43:08
			Example	44:38
		Cramer's Rule		52:22
			Introduction to Cramer's Rule and Example	52:23
III. Vectors in Rn
	Vectors in the Plane			46:54
		Intro		0:00
		Vectors in the Plane		0:38
			Vectors in the Plane	0:39
			Example 1	8:25
			Example 2	15:23
		Vector Addition and Scalar Multiplication		19:33
			Vector Addition	19:34
			Scalar Multiplication	24:08
			Example	26:25
		The Angle Between Two Vectors		29:33
			The Angle Between Two Vectors	29:34
			Example	33:54
		Properties of the Dot Product and Unit Vectors		38:17
			Properties of the Dot Product and Unit Vectors	38:18
			Defining Unit Vectors	40:01
			2 Very Important Unit Vectors	41:56
	n-Vector			52:44
		Intro		0:00
		n-Vectors		0:58
			4-Vector	0:59
			7-Vector	1:50
			Vector Addition	2:43
			Scalar Multiplication	3:37
			Theorem: Part 1	4:24
			Theorem: Part 2	11:38
			Right and Left Handed Coordinate System	14:19
			Projection of a Point Onto a Coordinate Line/Plane	17:20
			Example	21:27
			Cauchy-Schwarz Inequality	24:56
			Triangle Inequality	36:29
			Unit Vector	40:34
		Vectors and Dot Products		44:23
			Orthogonal Vectors	44:24
			Cauchy-Schwarz Inequality	45:04
			Triangle Inequality	45:21
			Example 1	45:40
			Example 2	48:16
	Linear Transformation			48:53
		Intro		0:00
		Introduction to Linear Transformations		0:44
			Introduction to Linear Transformations	0:45
			Example 1	9:01
			Example 2	11:33
			Definition of Linear Mapping	14:13
			Example 3	22:31
			Example 4	26:07
			Example 5	30:36
		Examples		36:12
			Projection Mapping	36:13
			Images, Range, and Linear Mapping	38:33
			Example of Linear Transformation	42:02
	Linear Transformations, Part II			34:08
		Intro		0:00
		Linear Transformations		1:29
			Linear Transformations	1:30
			Theorem 1	7:15
			Theorem 2	9:20
			Example 1: Find L (-3, 4, 2)	11:17
			Example 2: Is It Linear?	17:11
			Theorem 3	25:57
			Example 3: Finding the Standard Matrix	29:09
	Lines and Planes			37:54
		Intro		0:00
		Lines and Plane		0:36
			Example 1	0:37
			Example 2	7:07
			Lines in IR3	9:53
			Parametric Equations	14:58
			Example 3	17:26
			Example 4	20:11
			Planes in IR3	25:19
			Example 5	31:12
			Example 6	34:18
IV. Real Vector Spaces
	Vector Spaces			42:19
		Intro		0:00
		Vector Spaces		3:43
			Definition of Vector Spaces	3:44
			Vector Spaces 1	5:19
			Vector Spaces 2	9:34
			Real Vector Space and Complex Vector Space	14:01
			Example 1	15:59
			Example 2	18:42
		Examples		26:22
			More Examples	26:23
		Properties of Vector Spaces		32:53
			Properties of Vector Spaces Overview	32:54
			Property A	34:31
			Property B	36:09
			Property C	36:38
			Property D	37:54
			Property F	39:00
	Subspaces			43:37
		Intro		0:00
		Subspaces		0:47
			Defining Subspaces	0:48
			Example 1	3:08
			Example 2	3:49
			Theorem	7:26
			Example 3	9:11
			Example 4	12:30
			Example 5	16:05
		Linear Combinations		23:27
			Definition 1	23:28
			Example 1	25:24
			Definition 2	29:49
			Example 2	31:34
			Theorem	32:42
			Example 3	34:00
	Spanning Set for a Vector Space			33:15
		Intro		0:00
		A Spanning Set for a Vector Space		1:10
			A Spanning Set for a Vector Space	1:11
			Procedure to Check if a Set of Vectors Spans a Vector Space	3:38
			Example 1	6:50
			Example 2	14:28
			Example 3	21:06
			Example 4	22:15
	Linear Independence			17:20
		Intro		0:00
		Linear Independence		0:32
			Definition	0:39
			Meaning	3:00
			Procedure for Determining if a Given List of Vectors is Linear Independence or Linear Dependence	5:00
			Example 1	7:21
			Example 2	10:20
	Basis & Dimension			31:20
		Intro		0:00
		Basis and Dimension		0:23
			Definition	0:24
			Example 1	3:30
			Example 2: Part A	4:00
			Example 2: Part B	6:53
			Theorem 1	9:40
			Theorem 2	11:32
			Procedure for Finding a Subset of S that is a Basis for Span S	14:20
			Example 3	16:38
			Theorem 3	21:08
			Example 4	25:27
	Homogeneous Systems			24:45
		Intro		0:00
		Homogeneous Systems		0:51
			Homogeneous Systems	0:52
			Procedure for Finding a Basis for the Null Space of Ax = 0	2:56
			Example 1	7:39
			Example 2	18:03
			Relationship Between Homogeneous and Non-Homogeneous Systems	19:47
	Rank of a Matrix, Part I			35:03
		Intro		0:00
		Rank of a Matrix		1:47
			Definition	1:48
			Theorem 1	8:14
			Example 1	9:38
			Defining Row and Column Rank	16:53
			If We Want a Basis for Span S Consisting of Vectors From S	22:00
			If We want a Basis for Span S Consisting of Vectors Not Necessarily in S	24:07
			Example 2: Part A	26:44
			Example 2: Part B	32:10
	Rank of a Matrix, Part II			29:26
		Intro		0:00
		Rank of a Matrix		0:17
			Example 1: Part A	0:18
			Example 1: Part B	5:58
			Rank of a Matrix Review: Rows, Columns, and Row Rank	8:22
			Procedure for Computing the Rank of a Matrix	14:36
			Theorem 1: Rank + Nullity = n	16:19
			Example 2	17:48
			Rank & Singularity	20:09
			Example 3	21:08
			Theorem 2	23:25
		List of Non-Singular Equivalences		24:24
			List of Non-Singular Equivalences	24:25
	Coordinates of a Vector			27:03
		Intro		0:00
		Coordinates of a Vector		1:07
			Coordinates of a Vector	1:08
			Example 1	8:35
			Example 2	15:28
			Example 3: Part A	19:15
			Example 3: Part B	22:26
	Change of Basis & Transition Matrices			33:47
		Intro		0:00
		Change of Basis & Transition Matrices		0:56
			Change of Basis & Transition Matrices	0:57
			Example 1	10:44
			Example 2	20:44
			Theorem	23:37
			Example 3: Part A	26:21
			Example 3: Part B	32:05
	Orthonormal Bases in n-Space			32:53
		Intro		0:00
		Orthonormal Bases in n-Space		1:02
			Orthonormal Bases in n-Space: Definition	1:03
			Example 1	4:31
			Theorem 1	6:55
			Theorem 2	8:00
			Theorem 3	9:04
			Example 2	10:07
			Theorem 2	13:54
			Procedure for Constructing an O/N Basis	16:11
			Example 3	21:42
	Orthogonal Complements, Part I			21:27
		Intro		0:00
		Orthogonal Complements		0:19
			Definition	0:20
			Theorem 1	5:36
			Example 1	6:58
			Theorem 2	13:26
			Theorem 3	15:06
			Example 2	18:20
	Orthogonal Complements, Part II			33:49
		Intro		0:00
		Relations Among the Four Fundamental Vector Spaces Associated with a Matrix A		2:16
			Four Spaces Associated With A (If A is m x n)	2:17
			Theorem	4:49
			Example 1	7:17
			Null Space and Column Space	10:48
		Projections and Applications		16:50
			Projections and Applications	16:51
			Projection Illustration	21:00
			Example 1	23:51
			Projection Illustration Review	30:15
V. Eigenvalues and Eigenvectors
	Eigenvalues and Eigenvectors			38:11
		Intro		0:00
		Eigenvalues and Eigenvectors		0:38
			Eigenvalues and Eigenvectors	0:39
			Definition 1	3:30
			Example 1	7:20
			Example 2	10:19
			Definition 2	21:15
			Example 3	23:41
			Theorem 1	26:32
			Theorem 2	27:56
			Example 4	29:14
			Review	34:32
	Similar Matrices & Diagonalization			29:55
		Intro		0:00
		Similar Matrices and Diagonalization		0:25
			Definition 1	0:26
			Example 1	2:00
			Properties	3:38
			Definition 2	4:57
			Theorem 1	6:12
			Example 3	9:37
			Theorem 2	12:40
			Example 4	19:12
			Example 5	20:55
			Procedure for Diagonalizing Matrix A: Step 1	24:21
			Procedure for Diagonalizing Matrix A: Step 2	25:04
			Procedure for Diagonalizing Matrix A: Step 3	25:38
			Procedure for Diagonalizing Matrix A: Step 4	27:02
	Diagonalization of Symmetric Matrices			30:14
		Intro		0:00
		Diagonalization of Symmetric Matrices		1:15
			Diagonalization of Symmetric Matrices	1:16
			Theorem 1	2:24
			Theorem 2	3:27
			Example 1	4:47
			Definition 1	6:44
			Example 2	8:15
			Theorem 3	10:28
			Theorem 4	12:31
			Example 3	18:00
VI. Linear Transformations
	Linear Mappings Revisited			24:05
		Intro		0:00
		Linear Mappings		2:08
			Definition	2:09
			Linear Operator	7:36
			Projection	8:48
			Dilation	9:40
			Contraction	10:07
			Reflection	10:26
			Rotation	11:06
			Example 1	13:00
			Theorem 1	18:16
			Theorem 2	19:20
	Kernel and Range of a Linear Map, Part I			24:05
		Intro		0:00
		Kernel and Range of a Linear Map		0:28
			Definition 1	0:29
			Example 1	4:36
			Example 2	8:12
			Definition 2	10:34
			Example 3	13:34
			Theorem 1	16:01
			Theorem 2	18:26
			Definition 3	21:11
			Theorem 3	24:28
	Kernel and Range of a Linear Map, Part II			25:54
		Intro		0:00
		Kernel and Range of a Linear Map		1:39
			Theorem 1	1:40
			Example 1: Part A	2:32
			Example 1: Part B	8:12
			Example 1: Part C	13:11
			Example 1: Part D	14:55
			Theorem 2	16:50
			Theorem 3	23:00
	Matrix of a Linear Map			33:21
		Intro		0:00
		Matrix of a Linear Map		0:11
			Theorem 1	1:24
			Procedure for Computing to Matrix: Step 1	7:10
			Procedure for Computing to Matrix: Step 2	8:58
			Procedure for Computing to Matrix: Step 3	9:50
			Matrix of a Linear Map: Property	10:41
			Example 1	14:07
			Example 2	18:12
			Example 3	24:31

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Mathematics: Linear Algebra

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