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Khan Academy
Linear Algebra
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1
Introduction to matrices
by Khan Academy
622,621 views
Learn more: http://www.khanacademy.org/video?v=xyAuNHPsq-g
What a matrix is. How to add and subtract them. -
2
Matrix multiplication (part 1)
by Khan Academy
487,055 views
Learn more: http://www.khanacademy.org/video?v=aKhhYguY0DQ
Multiplying two 2x2 matrices. -
3
Matrix multiplication (part 2)
by Khan Academy
270,399 views
Learn more: http://www.khanacademy.org/video?v=OAh573i_qn8
More on multiplying matrices. -
4
Inverse Matrix (part 1)
by Khan Academy
476,771 views
Learn more: http://www.khanacademy.org/video?v=iUQR0enP7RQ
Taking the inverse of a 2x2 matrix -
5
Inverting matrices (part 2)
by Khan Academy
253,876 views
Learn more: http://www.khanacademy.org/video?v=S4n-tQZnU6o
Inverting a 3x3 matrix -
6
Inverting Matrices (part 3)
by Khan Academy
181,096 views
Learn more: http://www.khanacademy.org/video?v=obts_JDS6_Q
Using Gauss-Jordan elimination to invert a 3x3 matrix. -
7
Matrices to solve a system of equations
by Khan Academy
376,933 views
Learn more: http://www.khanacademy.org/video?v=AUqeb9Z3y3k
Using the inverse of a matrix to solve a system of equations. -
8
Matrices to solve a vector combination problem
by Khan Academy
120,807 views
Learn more: http://www.khanacademy.org/video?v=gsNgdVdAT1o
Using matrices to figure out if some combination of 2 vectors can create a 3rd vector -
9
Singular Matrices
by Khan Academy
136,535 views
Learn more: http://www.khanacademy.org/video?v=UqyN7-tRS00
When and why you can't invert a matrix. -
10
3-variable linear equations (part 1)
by Khan Academy
95,835 views
Learn more: http://www.khanacademy.org/video?v=5tB7y_piK6o
Visual intuition of a 3-variable linear equation. -
11
Solving 3 Equations with 3 Unknowns
by Khan Academy
167,290 views
Learn more: http://www.khanacademy.org/video?v=woqq3Sls1d8
Solving 3 equations with 3 unknowns (old video from July 2008) -
13
Linear Algebra: Vector Examples
by Khan Academy
227,269 views
Learn more: http://www.khanacademy.org/video?v=r4bH66vYjss
Visually understanding basic vector operations -
14
Linear Algebra: Parametric Representations of Lines
by Khan Academy
145,881 views
Learn more: http://www.khanacademy.org/video?v=hWhs2cIj7Cw
Linear Algebra: Parametric Representations of Lines in R2 and R3 -
15
Linear Combinations and Span
by Khan Academy
220,150 views
Learn more: http://www.khanacademy.org/video?v=Qm_OS-8COwU
Understanding linear combinations and spans of vectors -
16
Linear Algebra: Introduction to Linear Independence
by Khan Academy
172,505 views
Learn more: http://www.khanacademy.org/video?v=CrV1xCWdY-g
Linear Algebra: Introduction to linear dependence and independence -
17
More on linear independence
by Khan Academy
105,445 views
Learn more: http://www.khanacademy.org/video?v=Alhcv5d_XOs
More examples determining linear dependence or independence. -
18
Span and Linear Independence Example
by Khan Academy
142,729 views
Learn more: http://www.khanacademy.org/video?v=9kW6zFK5E5c
Determining whether 3 vectors are linearly independent and/or span R3 -
19
Linear Subspaces
by Khan Academy
218,624 views
Learn more: http://www.khanacademy.org/video?v=pMFv6liWK4M
Introduction to linear subspaces of Rn -
20
Linear Algebra: Basis of a Subspace
by Khan Academy
227,874 views
Learn more: http://www.khanacademy.org/video?v=zntNi3-ybfQ
Understanding the definition of a basis of a subspace -
21
Vector Dot Product and Vector Length
by Khan Academy
122,585 views
Learn more: http://www.khanacademy.org/video?v=WNuIhXo39_k
Definitions of the vector dot product and vector length -
22
Proving Vector Dot Product Properties
by Khan Academy
68,033 views
Learn more: http://www.khanacademy.org/video?v=rVQ3G9epCjw
Proving the "associative", "distributive" and "commutative" properties for vector dot products. -
23
Proof of the Cauchy-Schwarz Inequality
by Khan Academy
97,770 views
Learn more: http://www.khanacademy.org/video?v=r2PogGDl8_U
Proof of the Cauchy-Schwarz Inequality -
24
Linear Algebra: Vector Triangle Inequality
by Khan Academy
52,407 views
Learn more: http://www.khanacademy.org/video?v=PsNidCBr5II
Proving the triangle inequality for vectors in Rn -
25
Defining the angle between vectors
by Khan Academy
76,150 views
Learn more: http://www.khanacademy.org/video?v=5AWob_z74Ks
Introducing the idea of an angle between two vectors -
26
Defining a plane in R3 with a point and normal vector
by Khan Academy
69,254 views
Learn more: http://www.khanacademy.org/video?v=UJxgcVaNTqY
Determining the equation for a plane in R3 using a point on the plane and a normal vector -
27
Linear Algebra: Cross Product Introduction
by Khan Academy
71,354 views
Learn more: http://www.khanacademy.org/video?v=pJzmiywagfY
Introduction to the cross product -
28
Proof: Relationship between cross product and sin of angle
by Khan Academy
37,889 views
Learn more: http://www.khanacademy.org/video?v=7MKA2QlKvHc
Proof: Relationship between the cross product and sin of angle between vectors -
29
Dot and Cross Product Comparison/Intuition
by Khan Academy
54,118 views
Learn more: http://www.khanacademy.org/video?v=tdwFdzVqito
Dot and Cross Product Comparison/Intuition -
30
Matrices: Reduced Row Echelon Form 1
by Khan Academy
210,977 views
Learn more: http://www.khanacademy.org/video?v=L0CmbneYETs
Solving a system of linear equations by putting an augmented matrix into reduced row echelon form -
31
Matrices: Reduced Row Echelon Form 2
by Khan Academy
76,053 views
Learn more: http://www.khanacademy.org/video?v=lP1DGtZ8Wys
Another example of solving a system of linear equations by putting an augmented matrix into reduced row echelon form -
32
Matrices: Reduced Row Echelon Form 3
by Khan Academy
57,436 views
Learn more: http://www.khanacademy.org/video?v=JVDrlTdzxiI
And another example of solving a system of linear equations by putting an augmented matrix into reduced row echelon form -
33
Matrix Vector Products
by Khan Academy
54,155 views
Learn more: http://www.khanacademy.org/video?v=7Mo4S2wyMg4
Defining and understanding what it means to take the product of a matrix and a vector -
34
Introduction to the Null Space of a Matrix
by Khan Academy
122,202 views
Learn more: http://www.khanacademy.org/video?v=jCwRV1QL_Xs
Showing that the Null Space of a Matrix is a valid Subspace -
35
Null Space 2: Calculating the null space of a matrix
by Khan Academy
129,990 views
Learn more: http://www.khanacademy.org/video?v=qvyboGryeA8
Calculating the null space of a matrix -
36
Null Space 3: Relation to Linear Independence
by Khan Academy
56,771 views
Learn more: http://www.khanacademy.org/video?v=vFctYRhK2M0
Understanding how the null space of a matrix relates to the linear independence of its column vectors -
37
Column Space of a Matrix
by Khan Academy
73,704 views
Learn more: http://www.khanacademy.org/video?v=st6D5OdFV9M
Introduction to the column space of a matrix -
38
Null Space and Column Space Basis
by Khan Academy
116,104 views
Learn more: http://www.khanacademy.org/video?v=_uTAdf_AsfQ
Figuring out the null space and a basis of a column space for a matrix -
39
Visualizing a Column Space as a Plane in R3
by Khan Academy
29,672 views
Learn more: http://www.khanacademy.org/video?v=EGNlXtjYABw
Determining the planar equation for a column space in R3 -
40
Proof: Any subspace basis has same number of elements
by Khan Academy
25,019 views
Learn more: http://www.khanacademy.org/video?v=Zn2K8UIT8r4
Proof: Any subspace basis has same number of elements -
41
Dimension of the Null Space or Nullity
by Khan Academy
62,474 views
Learn more: http://www.khanacademy.org/video?v=abYAUqs_n6I
Dimension of the Null Space or Nullity -
42
Dimension of the Column Space or Rank
by Khan Academy
70,673 views
Learn more: http://www.khanacademy.org/video?v=JUgrBkPteTg
Dimension of the Column Space or Rank -
43
Showing relation between basis cols and pivot cols
by Khan Academy
22,408 views
Learn more: http://www.khanacademy.org/video?v=BfVjTOjvI30
Showing that linear independence of pivot columns implies linear independence of the corresponding columns in the original equation -
44
Showing that the candidate basis does span C(A)
by Khan Academy
17,192 views
Learn more: http://www.khanacademy.org/video?v=CkQOCnLWPUA
Showing that just the columns of A associated with the pivot columns of rref(A) do indeed span C(A). -
45
A more formal understanding of functions
by Khan Academy
81,888 views
Learn more: http://www.khanacademy.org/video?v=BQMyeQOLvpg
A more formal understanding of functions -
46
Vector Transformations
by Khan Academy
83,643 views
Learn more: http://www.khanacademy.org/video?v=gAbadNuQEjI
Introduction to the notion of vector transformations -
47
Linear Transformations
by Khan Academy
196,103 views
Learn more: http://www.khanacademy.org/video?v=4PCktDZJH8E
Introduction to linear transformations -
48
Matrix Vector Products as Linear Transformations
by Khan Academy
75,330 views
Learn more: http://www.khanacademy.org/video?v=ondmopWLiEg
Matrix Vector Products as Linear Transformations -
49
Linear Transformations as Matrix Vector Products
by Khan Academy
94,621 views
Learn more: http://www.khanacademy.org/video?v=PErhLkQcpZ8
Showing how ANY linear transformation can be represented as a matrix vector product -
50
Image of a subset under a transformation
by Khan Academy
38,187 views
Learn more: http://www.khanacademy.org/video?v=MIAmN5kgp3k
Exploring what happens to a subset of the domain under a transformation -
51
im(T): Image of a Transformation
by Khan Academy
45,400 views
Learn more: http://www.khanacademy.org/video?v=hZ827mfh1Jo
Showing that the image of a subspace under a transformation is also a subspace. Definition of the image of a Transformation. -
52
Preimage of a set
by Khan Academy
21,400 views
Learn more: http://www.khanacademy.org/video?v=zVsGtU8lIWs
Definition of preimage of a set -
53
Preimage and Kernel Example
by Khan Academy
45,495 views
Learn more: http://www.khanacademy.org/video?v=6yrPU8rYOhs
Example involving the preimage of a set under a transformation. Definition of kernel of a transformation. -
54
Sums and Scalar Multiples of Linear Transformations
by Khan Academy
18,668 views
Learn more: http://www.khanacademy.org/video?v=wHuY97vss18
Sums and Scalar Multiples of Linear Transformations. Definitions of matrix addition and scalar multiplication. -
55
More on Matrix Addition and Scalar Multiplication
by Khan Academy
21,005 views
Learn more: http://www.khanacademy.org/video?v=Z19-ObbmHJE
More on Matrix Addition and Scalar Multiplication -
56
Linear Transformation Examples: Scaling and Reflections
by Khan Academy
61,015 views
Learn more: http://www.khanacademy.org/video?v=qkfODKmZ-x4
Creating scaling and reflection transformation matrices (which are diagonal) -
57
Linear Transformation Examples: Rotations in R2
by Khan Academy
57,238 views
Learn more: http://www.khanacademy.org/video?v=lPWfIq5DzqI
Linear Transformation Examples: Rotations in R2 -
58
Rotation in R3 around the X-axis
by Khan Academy
35,036 views
Learn more: http://www.khanacademy.org/video?v=gkyuLPzfDV0
Construction a rotation transformation in R3 -
59
Unit Vectors
by Khan Academy
69,043 views
Learn more: http://www.khanacademy.org/video?v=lQn7fksaDq0
What unit vectors are and how to construct them -
60
Introduction to Projections
by Khan Academy
58,020 views
Learn more: http://www.khanacademy.org/video?v=27vT-NWuw0M
Determining the projection of a vector on s line -
61
Expressing a Projection on to a line as a Matrix Vector prod
by Khan Academy
29,453 views
Learn more: http://www.khanacademy.org/video?v=JK-8XNIoAkI
Expressing a Projection on to a line as a Matrix Vector prod -
62
Compositions of Linear Transformations 1
by Khan Academy
31,204 views
Learn more: http://www.khanacademy.org/video?v=f_DTiXZpb8M
Introduction to compositions of Linear Transformations -
63
Compositions of Linear Transformations 2
by Khan Academy
15,949 views
Learn more: http://www.khanacademy.org/video?v=BuqcKpe5ZQs
Providing the motivation for definition of matrix products -
64
Linear Algebra: Matrix Product Examples
by Khan Academy
24,848 views
Learn more: http://www.khanacademy.org/video?v=x1z0hOyjapU
Example of taking the product of two matrices -
65
Matrix Product Associativity
by Khan Academy
10,961 views
Learn more: http://www.khanacademy.org/video?v=Hhc96U_HvQE
Showing that matrix products are associative -
66
Distributive Property of Matrix Products
by Khan Academy
11,108 views
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Showing that matrix products exhibit the distributive property -
67
Linear Algebra: Introduction to the inverse of a function
by Khan Academy
51,920 views
Learn more: http://www.khanacademy.org/video?v=-eAzhBZgq28
Introduction to the inverse of a function -
68
Proof: Invertibility implies a unique solution to f(x)=y
by Khan Academy
17,658 views
Learn more: http://www.khanacademy.org/video?v=7GEUgRcnfVE
Proof: Invertibility implies a unique solution to f(x)=y for all y in co-domain of f. -
69
Surjective (onto) and Injective (one-to-one) functions
by Khan Academy
90,136 views
Learn more: http://www.khanacademy.org/video?v=xKNX8BUWR0g
Introduction to surjective and injective functions -
70
Relating invertibility to being onto and one-to-one
by Khan Academy
20,183 views
Learn more: http://www.khanacademy.org/video?v=QIU1daMN8fw
Relating invertibility to being onto (surjective) and one-to-one (injective) -
71
Determining whether a transformation is onto
by Khan Academy
27,597 views
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Determining whether a transformation is onto -
72
Linear Algebra: Exploring the solution set of Ax=b
by Khan Academy
38,018 views
Learn more: http://www.khanacademy.org/video?v=1PsNIzUJPkc
Exploring the solution set of Ax=b (non homogeneous equations) -
73
Linear Algebra: Matrix condition for one-to-one trans
by Khan Academy
25,699 views
Learn more: http://www.khanacademy.org/video?v=M3FuL9qKTBs
Showing that the rank of the of an mxn transformation matrix has to be an for the transformation to be one-to-one (injective) -
74
Linear Algebra: Simplifying conditions for invertibility
by Khan Academy
15,892 views
Learn more: http://www.khanacademy.org/video?v=Yz2OosyMTmY
Showing that a transformation is invertible if and only if rref(A) is equal to the identity matrix -
75
Linear Algebra: Showing that Inverses are Linear
by Khan Academy
10,583 views
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Showing that inverse transformations are also linear -
76
Linear Algebra: Deriving a method for determining inverses
by Khan Academy
24,559 views
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Determining a method for constructing inverse transformation matrices -
77
Linear Algebra: Example of Finding Matrix Inverse
by Khan Academy
44,394 views
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Example of calculating the inverse of a matrix -
78
Linear Algebra: Formula for 2x2 inverse
by Khan Academy
28,554 views
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Figuring out the formula for a 2x2 matrix. Defining the determinant. -
79
Linear Algebra: 3x3 Determinant
by Khan Academy
85,753 views
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Determinants: Finding the determinant of a 3x3 matrix -
80
Linear Algebra: nxn Determinant
by Khan Academy
68,503 views
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Defining the determinant for nxn matrices. An exampled of a 4x4 determinant. -
81
Linear Algebra: Determinants along other rows/cols
by Khan Academy
22,736 views
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Finding the determinant by going along other rows or columns -
82
Linear Algebra: Rule of Sarrus of Determinants
by Khan Academy
26,238 views
Learn more: http://www.khanacademy.org/video?v=4xFIi0JF2AM
A alternative "short cut" for calculating 3x3 determinants (Rule of Sarrus) -
83
Linear Algebra: Determinant when row multiplied by scalar
by Khan Academy
16,921 views
Learn more: http://www.khanacademy.org/video?v=32rdijPB-rA
The determinant when a row is multiplied by a scalar -
84
Linear Algebra: (correction) scalar muliplication of row
by Khan Academy
8,899 views
Learn more: http://www.khanacademy.org/video?v=WY6QG0mZ7uQ
Correction of last video showing that the determinant when one row is multiplied by a scalar is equal to the scalar times the determinant -
85
Linear Algebra: Determinant when row is added
by Khan Academy
12,408 views
Learn more: http://www.khanacademy.org/video?v=VrB3LaSD_uo
The determinant when one matrix has a row that is the sum of the rows of other matrices (and every other term is identical in the 3 matrices) -
86
Linear Algebra: Duplicate Row Determinant
by Khan Academy
8,972 views
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Determinant of a matrix with duplicate rows -
87
Linear Algebra: Determinant after row operations
by Khan Academy
15,960 views
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What happens to the determinant when we perform a row operation -
88
Linear Algebra: Upper Triangular Determinant
by Khan Academy
26,858 views
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The determinant of an upper triangular matrix -
89
Linear Algebra: Simpler 4x4 determinant
by Khan Academy
92,881 views
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Calculating a 4x4 determinant by putting in in upper triangular form first. -
90
Linear Algebra: Determinant and area of a parallelogram
by Khan Academy
23,977 views
Learn more: http://www.khanacademy.org/video?v=n-S63_goDFg
Realizing that the determinant of a 2x2 matrix is equal to the area of the parallelogram defined by the column vectors of the matrix -
91
Linear Algebra: Determinant as Scaling Factor
by Khan Academy
15,233 views
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Viewing the determinant of the transformation matrix as a scaling factor of regions -
92
Linear Algebra: Transpose of a Matrix
by Khan Academy
54,141 views
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Transpose of a matrix -
93
Linear Algebra: Determinant of Transpose
by Khan Academy
20,885 views
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Proof by induction that transposing a matrix does not change its determinant -
94
Linear Algebra: Transpose of a Matrix Product
by Khan Academy
13,272 views
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Taking the transpose of the product of two matrices -
95
Linear Algebra: Transposes of sums and inverses
by Khan Academy
10,449 views
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Transposes of sums and inverses -
96
Linear Algebra: Transpose of a Vector
by Khan Academy
15,115 views
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Transpose of a column vector. Matrix-matrix products using vectors -
97
Linear Algebra: Rowspace and Left Nullspace
by Khan Academy
25,283 views
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Rowspace and Left Nullspace -
98
Lin Alg: Visualizations of Left Nullspace and Rowspace
by Khan Academy
10,151 views
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Relationship between left nullspace, rowspace, column space and nullspace. -
99
Linear Algebra: Orthogonal Complements
by Khan Academy
58,003 views
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Orthogonal Complements as subspaces. -
100
Linear Algebra: Rank(A) = Rank(transpose of A)
by Khan Academy
31,010 views
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Rank(A) = Rank(transpose of A) -
101
Linear Algebra: dim(V) + dim(orthogonoal complelent of V)=n
by Khan Academy
15,790 views
Learn more: http://www.khanacademy.org/video?v=pGhBfkBqok0
Showing that if V is a subspace of Rn, then dim(V) + dim(V's orthogonal complement) = n -
102
Lin Alg: Representing vectors in Rn using subspace members
by Khan Academy
12,503 views
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Showing that any member of Rn can be represented as a unique sum of a vector in subspace V and a vector in the orthogonal complement of V. -
103
Lin Alg: Orthogonal Complement of the Orthogonal Complement
by Khan Academy
14,638 views
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Finding that the orthogonal complement of the orthogonal complement of V is V -
104
Lin Alg: Orthogonal Complement of the Nullspace
by Khan Academy
11,667 views
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The orthogonal complement of the nullspace and left nullspace. -
105
Lin Alg: Unique rowspace solution to Ax=b
by Khan Academy
10,097 views
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Showing that, for any b that is in the column space of A, there is a unique member of the row space that is the "smallest" solution to Ax=b -
106
Linear Alg: Rowspace Solution to Ax=b example
by Khan Academy
13,107 views
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Visualizing the rowspace solution to Ax=b -
107
Linear Alg: Rowspace Solution to Ax=b example
by Khan Academy
13,107 views
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Visualizing the rowspace solution to Ax=b -
108
Lin Alg: Showing that A-transpose x A is invertible
by Khan Academy
13,359 views
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Showing that (transpose of A)(A) is invertible if A has linearly independent columns -
109
Linear Algebra: Projections onto Subspaces
by Khan Academy
30,672 views
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Projections onto subspaces -
110
Linear Alg: Visualizing a projection onto a plane
by Khan Academy
16,190 views
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Visualizing a projection onto a plane. Showing that the old and new definitions of projections aren't that different. -
111
Lin Alg: A Projection onto a Subspace is a Linear Transforma
by Khan Academy
12,239 views
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Showing that a projection onto a subspace is a linear transformation -
112
Linear Algebra: Subspace Projection Matrix Example
by Khan Academy
25,220 views
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Example of a transformation matrix for a projection onto a subspace -
113
Lin Alg: Another Example of a Projection Matrix
by Khan Academy
13,300 views
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Figuring out the transformation matrix for a projection onto a subspace by figuring out the matrix for the projection onto the subspace's orthogonal complement first -
114
Linear Alg: Projection is closest vector in subspace
by Khan Academy
11,216 views
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Showing that the projection of x onto a subspace is the closest vector in the subspace to x -
115
Linear Algebra: Least Squares Approximation
by Khan Academy
55,333 views
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The least squares approximation for otherwise unsolvable equations -
116
Linear Algebra: Least Squares Examples
by Khan Academy
50,045 views
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An example using the least squares solution to an unsolvable system -
117
Linear Algebra: Another Least Squares Example
by Khan Academy
25,961 views
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Using least squares approximation to fit a line to points -
118
Linear Algebra: Coordinates with Respect to a Basis
by Khan Academy
42,924 views
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Understanding alternate coordinate systems -
119
Linear Algebra: Change of Basis Matrix
by Khan Academy
84,806 views
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Using a change of basis matrix to get us from one coordinate system to another. -
120
Lin Alg: Invertible Change of Basis Matrix
by Khan Academy
20,392 views
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Using an invertible change of basis matrix to go between different coordinate systems -
121
Lin Alg: Transformation Matrix with Respect to a Basis
by Khan Academy
55,734 views
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Finding the transformation matrix with respect to a non-standard basis -
122
Lin Alg: Alternate Basis Tranformation Matrix Example
by Khan Academy
21,099 views
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Example of finding the transformation matrix for an alternate basis -
123
Lin Alg: Alternate Basis Tranformation Matrix Example Part 2
by Khan Academy
12,109 views
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Showing that the transformation matrix with respect to basis B actually works. Brief point on why someone would want to operate in a different basis to begin with. -
124
Lin Alg: Changing coordinate systems to help find a transformation matrix
by Khan Academy
24,283 views
Learn more: http://www.khanacademy.org/video?v=K_na_j1Msfg
Changing our coordinate system to find the transformation matrix with respect to standard coordinates -
125
Linear Algebra: Introduction to Orthonormal Bases
by Khan Academy
51,056 views
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Looking at sets and bases that are orthonormal -- or where all the vectors have length 1 and are orthogonal to each other. -
126
Linear Algebra: Coordinates with respect to orthonormal bases
by Khan Academy
17,904 views
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Seeing that orthonormal bases make for good coordinate systems -
127
Lin Alg: Projections onto subspaces with orthonormal bases
by Khan Academy
15,505 views
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Projections onto subspaces with orthonormal bases -
128
Lin Alg: Finding projection onto subspace with orthonormal basis example
by Khan Academy
15,865 views
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Example of finding the transformation matrix for the projection onto a subspace with an orthonormal basis -
129
Lin Alg: Example using orthogonal change-of-basis matrix to find transformation matrix
by Khan Academy
16,495 views
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Example using orthogonal change-of-basis matrix to find transformation matrix -
130
Lin Alg: Orthogonal matrices preserve angles and lengths
by Khan Academy
16,944 views
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Showing that orthogonal matrices preserve angles and lengths -
131
Linear Algebra: The Gram-Schmidt Process
by Khan Academy
74,541 views
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Finding an orthonormal basis for a subspace using the Gram-Schmidt Process -
132
Linear Algebra: Gram-Schmidt Process Example
by Khan Academy
48,950 views
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Using Gram-Schmidt to find an orthonormal basis for a plane in R3 -
133
Linear Algebra: Gram-Schmidt example with 3 basis vectors
by Khan Academy
39,679 views
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Gram-Schmidt example with 3 basis vectors -
134
Linear Algebra: Introduction to Eigenvalues and Eigenvectors
by Khan Academy
241,650 views
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What eigenvectors and eigenvalues are and why they are interesting -
135
Linear Algebra: Proof of formula for determining Eigenvalues
by Khan Academy
66,667 views
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Proof of formula for determining Eigenvalues -
136
Linear Algebra: Example solving for the eigenvalues of a 2x2 matrix
by Khan Academy
122,147 views
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Example solving for the eigenvalues of a 2x2 matrix -
137
Linear Algebra: Finding Eigenvectors and Eigenspaces example
by Khan Academy
197,007 views
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Finding the eigenvectors and eigenspaces of a 2x2 matrix -
138
Linear Algebra: Eigenvalues of a 3x3 matrix
by Khan Academy
196,808 views
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Determining the eigenvalues of a 3x3 matrix -
139
Linear Algebra: Eigenvectors and Eigenspaces for a 3x3 matrix
by Khan Academy
142,706 views
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Eigenvectors and eigenspaces for a 3x3 matrix -
140
Linear Algebra: Showing that an eigenbasis makes for good coordinate systems
by Khan Academy
33,915 views
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Showing that an eigenbasis makes for good coordinate systems -
141
Vector Triple Product Expansion (very optional)
by Khan Academy
25,975 views
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A shortcut for having to evaluate the cross product of three vectors -
142
Normal vector from plane equation
by Khan Academy
64,827 views
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Figuring out a normal vector to a plane from its equation -
143
Point distance to plane
by Khan Academy
41,953 views
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Distance between a point and a plane in three dimensions -
144
Distance Between Planes
by Khan Academy
36,695 views
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2010 IIT JEE Paper 1 Problem 51 Distance Between Planes
