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Linear Algebra - Khan Academy FREE

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Matrices, vectors, vector spaces, transformations. Covers all topics in a first year college linear algebra course. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra.

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Lessons
1) Introduction to matrices What a matrix is. How to add and subtract them. 594 views watch video
2) Matrix multiplication (part 1) Multiplying two 2x2 matrices. 532 views watch video
3) Matrix multiplication (part 2) More on multiplying matrices. 432 views watch video
4) Idea behind inverting a 2x2 matrix \| Matrices \| Precalculus What the inverse of a matrix is. Examples of inverting a 2x2 matrix. Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/preca ... 503 views watch video
5) Classic video on inverting a 3x3 matrix part 1 \| Matrices \| Precalculus Inverting a 3x3 matrix Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/preca ... 537 views watch video
6) Classic video on inverting a 3x3 matrix part 2 \| Matrices \| Precalculus Using Gauss-Jordan elimination to invert a 3x3 matrix. Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/preca ... 572 views watch video
7) Matrices to solve a system of equations \| Matrices \| Precalculus Using the inverse of a matrix to solve a system of equations. 519 views watch video
8) Matrices to solve a vector combination problem \| Matrices \| Precalculus Using matrices to figure out if some combination of 2 vectors can create a 3rd vector 572 views watch video
9) Singular matrices \| Matrices \| Precalculus When and why you can't invert a matrix. Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/preca ... 479 views watch video
10) 3-variable linear equations (part 1) Visual intuition of a 3-variable linear equation. 464 views watch video
11) Solving 3 Equations with 3 Unknowns Solving 3 equations with 3 unknowns (old video from July 2008) 520 views watch video
12) Vector examples \| Vectors and spaces \| Linear Algebra Visually understanding basic vector operations Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/linear- ... 525 views watch video
13) Parametric representations of lines \| Vectors and spaces \| Linear Algebra Parametric Representations of Lines in R2 and R3 425 views watch video
14) Linear combinations and span \| Vectors and spaces \| Linear Algebra Understanding linear combinations and spans of vectors 476 views watch video
15) Introduction to linear independence \| Vectors and spaces \| Linear Algebra Introduction to linear dependence and independence 520 views watch video
16) More on linear independence \| Vectors and spaces \| Linear Algebra More examples determining linear dependence or independence. 537 views watch video
17) Span and linear independence example \| Vectors and spaces \| Linear Algebra Determining whether 3 vectors are linearly independent and/or span R3 466 views watch video
18) Linear subspaces \| Vectors and spaces \| Linear Algebra Introduction to linear subspaces of Rn 632 views watch video
19) Basis of a subspace \| Vectors and spaces \| Linear Algebra Understanding the definition of a basis of a subspace 481 views watch video
20) Vector dot product and vector length \| Vectors and spaces \| Linear Algebra Definitions of the vector dot product and vector length 574 views watch video
21) Proving vector dot product properties \| Vectors and spaces \| Linear Algebra Proving the "associative", "distributive" and "commutative" properties for vector dot products. 528 views watch video
22) Proof of the Cauchy-Schwarz inequality \| Vectors and spaces \| Linear Algebra Proof of the Cauchy-Schwarz Inequality 744 views watch video
23) Vector triangle inequality \| Vectors and spaces \| Linear Algebra Proving the triangle inequality for vectors in Rn 492 views watch video
24) Defining the angle between vectors \| Vectors and spaces \| Linear Algebra Introducing the idea of an angle between two vectors 501 views watch video
25) Defining a plane in R3 with a point and normal vector \| Linear Algebra Determining the equation for a plane in R3 using a point on the plane and a normal vector 597 views watch video
26) Cross product introduction \| Vectors and spaces \| Linear Algebra Introduction to the cross product 563 views watch video
27) Proof: Relationship between cross product and sin of angle \| Linear Algebra Proof: Relationship between the cross product and sin of angle between vectors 567 views watch video
28) Dot and cross product comparison/intuition \| Vectors and spaces \| Linear Algebra Dot and Cross Product Comparison/Intuition 506 views watch video
29) Matrices: Reduced row echelon form 1 \| Vectors and spaces \| Linear Algebra Solving a system of linear equations by putting an augmented matrix into reduced row echelon form 549 views watch video
30) Matrices: Reduced row echelon form 2 \| Vectors and spaces \| Linear Algebra Another example of solving a system of linear equations by putting an augmented matrix into reduced row echelon form 683 views watch video
31) Matrices: Reduced row echelon form 3 \| Vectors and spaces \| Linear Algebra And another example of solving a system of linear equations by putting an augmented matrix into reduced row echelon form 546 views watch video
32) Matrix vector products \| Vectors and spaces \| Linear Algebra Defining and understanding what it means to take the product of a matrix and a vector 515 views watch video
33) Introduction to the null space of a matrix \| Vectors and spaces \| Linear Algebra Showing that the Null Space of a Matrix is a valid Subspace 624 views watch video
34) Null space 2: Calculating the null space of a matrix \| Linear Algebra Calculating the null space of a matrix 481 views watch video
35) Null space 3: Relation to linear independence \| Vectors and spaces \| Linear Algebra Understanding how the null space of a matrix relates to the linear independence of its column vectors 468 views watch video
36) Column space of a matrix \| Vectors and spaces \| Linear Algebra Introduction to the column space of a matrix 497 views watch video
37) Null space and column space basis \| Vectors and spaces \| Linear Algebra Figuring out the null space and a basis of a column space for a matrix 588 views watch video
38) Visualizing a column space as a plane in R3 \| Vectors and spaces \| Linear Algebra Determining the planar equation for a column space in R3 495 views watch video
39) Proof: Any subspace basis has same number of elements \| Linear Algebra Proof: Any subspace basis has same number of elements 506 views watch video
40) Dimension of the null space or nullity \| Vectors and spaces \| Linear Algebra Dimension of the Null Space or Nullity 505 views watch video
41) Dimension of the column space or rank \| Vectors and spaces \| Linear Algebra Dimension of the Column Space or Rank 575 views watch video
42) Showing relation between basis cols and pivot cols \| Linear Algebra Showing that linear independence of pivot columns implies linear independence of the corresponding columns in the original equation 583 views watch video
43) Showing that the candidate basis does span C(A) \| Vectors and spaces \| Linear Algebra Showing that just the columns of A associated with the pivot columns of rref(A) do indeed span C(A). 579 views watch video
44) A more formal understanding of functions \| Matrix transformations \| Linear Algebra A more formal understanding of functions 477 views watch video
45) Vector transformations \| Matrix transformations \| Linear Algebra Introduction to the notion of vector transformations 547 views watch video
46) Linear transformations \| Matrix transformations \| Linear Algebra Introduction to linear transformations 562 views watch video
47) Matrix vector products as linear transformations \| Linear Algebra Matrix Vector Products as Linear Transformations 507 views watch video
48) Linear transformations as matrix vector products \| Linear Algebra Showing how ANY linear transformation can be represented as a matrix vector product 481 views watch video
49) Image of a subset under a transformation \| Matrix transformations \| Linear Algebra Exploring what happens to a subset of the domain under a transformation 536 views watch video
50) im(T): Image of a transformation \| Matrix transformations \| Linear Algebra Showing that the image of a subspace under a transformation is also a subspace. Definition of the image of a Transformation. 599 views watch video
51) Preimage of a set \| Matrix transformations \| Linear Algebra Definition of preimage of a set 483 views watch video
52) Preimage and kernel example \| Matrix transformations \| Linear Algebra Example involving the preimage of a set under a transformation. Definition of kernel of a transformation. 515 views watch video
53) Sums and scalar multiples of linear transformations \| Linear Algebra Sums and Scalar Multiples of Linear Transformations. Definitions of matrix addition and scalar multiplication. 498 views watch video
54) More on matrix addition and scalar multiplication \| Linear Algebra More on Matrix Addition and Scalar Multiplication 601 views watch video
55) Linear transformation examples: Scaling and reflections \| Linear Algebra Creating scaling and reflection transformation matrices (which are diagonal) 556 views watch video
56) Linear transformation examples: Rotations in R2 \| Linear Algebra Linear Transformation Examples: Rotations in R2 417 views watch video
57) Rotation in R3 around the x-axis \| Matrix transformations \| Linear Algebra Construction a rotation transformation in R3 507 views watch video
58) Unit vectors \| Matrix transformations \| Linear Algebra What unit vectors are and how to construct them 441 views watch video
59) Introduction to projections \| Matrix transformations \| Linear Algebra Determining the projection of a vector on s line 446 views watch video
60) Expressing a projection on to a line as a matrix vector prod \| Linear Algebra Expressing a Projection on to a line as a Matrix Vector prod 467 views watch video
61) Compositions of linear transformations 1 \| Matrix transformations \| Linear Algebra Introduction to compositions of Linear Transformations 574 views watch video
62) Compositions of linear transformations 2 \| Matrix transformations \| Linear Algebra Providing the motivation for definition of matrix products 466 views watch video
63) Matrix product examples \| Matrix transformations \| Linear Algebra Example of taking the product of two matrices 521 views watch video
64) Matrix product associativity \| Matrix transformations \| Linear Algebra Showing that matrix products are associative 552 views watch video
65) Distributive property of matrix products \| Matrix transformations \| Linear Algebra Showing that matrix products are associative 405 views watch video
66) Introduction to the inverse of a function \| Matrix transformations \| Linear Algebra Introduction to the inverse of a function 437 views watch video
67) Proof: Invertibility implies a unique solution to f(x)=y \| Linear Algebra Proof: Invertibility implies a unique solution to f(x)=y for all y in co-domain of f. 472 views watch video
68) Surjective (onto) and injective (one-to-one) functions \| Linear Algebra Introduction to surjective and injective functions 500 views watch video
69) Relating invertibility to being onto and one-to-one \| Linear Algebra Relating invertibility to being onto (surjective) and one-to-one (injective) 547 views watch video
70) Determining whether a transformation is onto \| Linear Algebra Determining whether a transformation is onto 577 views watch video
71) Exploring the solution set of Ax = b \| Matrix transformations \| Linear Algebra Exploring the solution set of Ax=b (non homogeneous equations) 601 views watch video
72) Matrix condition for one-to-one trans \| Matrix transformations \| Linear Algebra Showing that the rank of the of an mxn transformation matrix has to be an for the transformation to be one-to-one (injective) 441 views watch video
73) Simplifying conditions for invertibility \| Matrix transformations \| Linear Algebra Showing that a transformation is invertible if and only if rref(A) is equal to the identity matrix 438 views watch video
74) Showing that inverses are linear \| Matrix transformations \| Linear Algebra Showing that inverse transformations are also linear 441 views watch video
75) Deriving a method for determining inverses \| Matrix transformations \| Linear Algebra Determining a method for constructing inverse transformation matrices 476 views watch video
76) Example of finding matrix inverse \| Matrix transformations \| Linear Algebra Example of calculating the inverse of a matrix 468 views watch video
77) Formula for 2x2 inverse \| Matrix transformations \| Linear Algebra Figuring out the formula for a 2x2 matrix. Defining the determinant. 436 views watch video
78) 3 x 3 determinant \| Matrix transformations \| Linear Algebra Determinants: Finding the determinant of a 3x3 matrix 531 views watch video
79) n x n determinant \| Matrix transformations \| Linear Algebra Defining the determinant for nxn matrices. An example of a 4x4 determinant. 537 views watch video
80) Determinants along other rows/cols \| Matrix transformations \| Linear Algebra Finding the determinant by going along other rows or columns 434 views watch video
81) Rule of Sarrus of determinants \| Matrix transformations \| Linear Algebra A alternative "short cut" for calculating 3x3 determinants (Rule of Sarrus) 592 views watch video
82) Determinant when row multiplied by scalar \| Matrix transformations \| Linear Algebra The determinant when a row is multiplied by a scalar 663 views watch video
83) (correction) scalar multiplication of row \| Matrix transformations \| Linear Algebra Correction of last video showing that the determinant when one row is multiplied by a scalar is equal to the scalar times the determinant 482 views watch video
84) Determinant when row is added \| Matrix transformations \| Linear Algebra 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) 567 views watch video
85) Duplicate row determinant \| Matrix transformations \| Linear Algebra Determinant of a matrix with duplicate rows 553 views watch video
86) Determinant after row operations \| Matrix transformations \| Linear Algebra What happens to the determinant when we perform a row operation 502 views watch video
87) Upper triangular determinant \| Matrix transformations \| Linear Algebra The determinant of an upper triangular matrix 468 views watch video
88) Simpler 4x4 determinant \| Matrix transformations \| Linear Algebra Calculating a 4x4 determinant by putting in in upper triangular form first. 467 views watch video
89) Determinant and area of a parallelogram \| Matrix transformations \| Linear Algebra Realizing that the determinant of a 2x2 matrix is equal to the area of the parallelogram defined by the column vectors of the matrix 559 views watch video
90) Determinant as scaling factor \| Matrix transformations \| Linear Algebra Viewing the determinant of the transformation matrix as a scaling factor of regions 484 views watch video
91) Transpose of a matrix \| Matrix transformations \| Linear Algebra Transpose of a matrix 569 views watch video
92) Determinant of transpose \| Matrix transformations \| Linear Algebra Proof by induction that transposing a matrix does not change its determinant 484 views watch video
93) Transpose of a matrix product \| Matrix transformations \| Linear Algebra Taking the transpose of the product of two matrices 455 views watch video
94) Transposes of sums and inverses \| Matrix transformations \| Linear Algebra Transposes of sums and inverses 461 views watch video
95) Transpose of a vector \| Matrix transformations \| Linear Algebra Transpose of a column vector. Matrix-matrix products using vectors 511 views watch video
96) Rowspace and left nullspace \| Matrix transformations \| Linear Algebra Rowspace and Left Nullspace 601 views watch video
97) Visualizations of left nullspace and rowspace \| Linear Algebra Relationship between left nullspace, rowspace, column space and nullspace. 457 views watch video
98) Orthogonal complements \| Alternate coordinate systems (bases) \| Linear Algebra Orthogonal Complements as subspaces. 506 views watch video
99) rank(a) = rank(transpose of a) \| Matrix transformations \| Linear Algebra Rank(A) = Rank(transpose of A) 413 views watch video
100) dim(v) + dim(orthogonal complement of v) = n \| Linear Algebra Showing that if V is a subspace of Rn, then dim(V) + dim(V's orthogonal complement) = n 456 views watch video

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Linear Algebra - Khan Academy FREE

Lessons

What a matrix is. How to add and subtract them.

Multiplying two 2x2 matrices.

More on multiplying matrices.

What the inverse of a matrix is. Examples of inverting a 2x2 matrix. Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/preca ...

Inverting a 3x3 matrix Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/preca ...

Using Gauss-Jordan elimination to invert a 3x3 matrix. Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/preca ...

Using the inverse of a matrix to solve a system of equations.

Using matrices to figure out if some combination of 2 vectors can create a 3rd vector

When and why you can't invert a matrix. Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/preca ...

Visual intuition of a 3-variable linear equation.

Solving 3 equations with 3 unknowns (old video from July 2008)

Visually understanding basic vector operations Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/linear- ...

Parametric Representations of Lines in R2 and R3

Understanding linear combinations and spans of vectors

Introduction to linear dependence and independence

More examples determining linear dependence or independence.

Determining whether 3 vectors are linearly independent and/or span R3

Introduction to linear subspaces of Rn

Understanding the definition of a basis of a subspace

Definitions of the vector dot product and vector length

Proving the "associative", "distributive" and "commutative" properties for vector dot products.

Proof of the Cauchy-Schwarz Inequality

Proving the triangle inequality for vectors in Rn

Introducing the idea of an angle between two vectors

Determining the equation for a plane in R3 using a point on the plane and a normal vector

Introduction to the cross product

Proof: Relationship between the cross product and sin of angle between vectors

Dot and Cross Product Comparison/Intuition

Solving a system of linear equations by putting an augmented matrix into reduced row echelon form

Another example of solving a system of linear equations by putting an augmented matrix into reduced row echelon form

And another example of solving a system of linear equations by putting an augmented matrix into reduced row echelon form

Defining and understanding what it means to take the product of a matrix and a vector

Showing that the Null Space of a Matrix is a valid Subspace

Calculating the null space of a matrix

Understanding how the null space of a matrix relates to the linear independence of its column vectors

Introduction to the column space of a matrix

Figuring out the null space and a basis of a column space for a matrix

Determining the planar equation for a column space in R3

Proof: Any subspace basis has same number of elements

Dimension of the Null Space or Nullity

Dimension of the Column Space or Rank

Showing that linear independence of pivot columns implies linear independence of the corresponding columns in the original equation

Showing that just the columns of A associated with the pivot columns of rref(A) do indeed span C(A).

A more formal understanding of functions

Introduction to the notion of vector transformations

Introduction to linear transformations

Matrix Vector Products as Linear Transformations

Showing how ANY linear transformation can be represented as a matrix vector product

Exploring what happens to a subset of the domain under a transformation

Showing that the image of a subspace under a transformation is also a subspace. Definition of the image of a Transformation.

Definition of preimage of a set

Example involving the preimage of a set under a transformation. Definition of kernel of a transformation.

Sums and Scalar Multiples of Linear Transformations. Definitions of matrix addition and scalar multiplication.

More on Matrix Addition and Scalar Multiplication

Creating scaling and reflection transformation matrices (which are diagonal)

Linear Transformation Examples: Rotations in R2

Construction a rotation transformation in R3

What unit vectors are and how to construct them

Determining the projection of a vector on s line

Expressing a Projection on to a line as a Matrix Vector prod

Introduction to compositions of Linear Transformations

Providing the motivation for definition of matrix products

Example of taking the product of two matrices

Showing that matrix products are associative

Showing that matrix products are associative

Introduction to the inverse of a function

Proof: Invertibility implies a unique solution to f(x)=y for all y in co-domain of f.

Introduction to surjective and injective functions

Relating invertibility to being onto (surjective) and one-to-one (injective)

Determining whether a transformation is onto

Exploring the solution set of Ax=b (non homogeneous equations)

Showing that the rank of the of an mxn transformation matrix has to be an for the transformation to be one-to-one (injective)

Showing that a transformation is invertible if and only if rref(A) is equal to the identity matrix

Showing that inverse transformations are also linear