Linear Algebra

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.

Curated by: Khan Academy (144 videos)

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Currently Playing: 3-variable linear equations (part 1)

Visual intuition of a 3-variable linear equation.

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Tracks in this Playlist

• Introduction to matrices
• Matrix multiplication (part 1)
• Matrix multiplication (part 2)
• Idea behind inverting a 2x2 matrix | Matrices | Precalculus | Khan Academy
• Classic video on inverting a 3x3 matrix part 1 | Matrices | Precalculus | Khan Academy
• Classic video on inverting a 3x3 matrix part 2 | Matrices | Precalculus | Khan Academy
• Matrices to solve a system of equations | Matrices | Precalculus | Khan Academy
• Matrices to solve a vector combination problem | Matrices | Precalculus | Khan Academy
• Singular matrices | Matrices | Precalculus | Khan Academy
• 3-variable linear equations (part 1)
• Solving 3 Equations with 3 Unknowns
• Linear Algebra: Introduction to Vectors
• Vector examples | Vectors and spaces | Linear Algebra | Khan Academy
• Parametric representations of lines | Vectors and spaces | Linear Algebra | Khan Academy
• Linear combinations and span | Vectors and spaces | Linear Algebra | Khan Academy
• Introduction to linear independence | Vectors and spaces | Linear Algebra | Khan Academy
• More on linear independence | Vectors and spaces | Linear Algebra | Khan Academy
• Span and linear independence example | Vectors and spaces | Linear Algebra | Khan Academy
• Linear subspaces | Vectors and spaces | Linear Algebra | Khan Academy
• Basis of a subspace | Vectors and spaces | Linear Algebra | Khan Academy
• Vector dot product and vector length | Vectors and spaces | Linear Algebra | Khan Academy
• Proving vector dot product properties | Vectors and spaces | Linear Algebra | Khan Academy
• Proof of the Cauchy-Schwarz inequality | Vectors and spaces | Linear Algebra | Khan Academy
• Vector triangle inequality | Vectors and spaces | Linear Algebra | Khan Academy
• Defining the angle between vectors | Vectors and spaces | Linear Algebra | Khan Academy
• Defining a plane in R3 with a point and normal vector | Linear Algebra | Khan Academy
• Cross product introduction | Vectors and spaces | Linear Algebra | Khan Academy
• Proof: Relationship between cross product and sin of angle | Linear Algebra | Khan Academy
• Dot and cross product comparison/intuition | Vectors and spaces | Linear Algebra | Khan Academy
• Matrices: Reduced row echelon form 1 | Vectors and spaces | Linear Algebra | Khan Academy
• Matrices: Reduced row echelon form 2 | Vectors and spaces | Linear Algebra | Khan Academy
• Matrices: Reduced row echelon form 3 | Vectors and spaces | Linear Algebra | Khan Academy
• Matrix vector products | Vectors and spaces | Linear Algebra | Khan Academy
• Introduction to the null space of a matrix | Vectors and spaces | Linear Algebra | Khan Academy
• Null space 2: Calculating the null space of a matrix | Linear Algebra | Khan Academy
• Null space 3: Relation to linear independence | Vectors and spaces | Linear Algebra | Khan Academy
• Column space of a matrix | Vectors and spaces | Linear Algebra | Khan Academy
• Null space and column space basis | Vectors and spaces | Linear Algebra | Khan Academy
• Visualizing a column space as a plane in R3 | Vectors and spaces | Linear Algebra | Khan Academy
• Proof: Any subspace basis has same number of elements | Linear Algebra | Khan Academy
• Dimension of the null space or nullity | Vectors and spaces | Linear Algebra | Khan Academy
• Dimension of the column space or rank | Vectors and spaces | Linear Algebra | Khan Academy
• Showing relation between basis cols and pivot cols | Linear Algebra | Khan Academy
• Showing that the candidate basis does span C(A) | Vectors and spaces | Linear Algebra | Khan Academy
• A more formal understanding of functions | Matrix transformations | Linear Algebra | Khan Academy
• Vector transformations | Matrix transformations | Linear Algebra | Khan Academy
• Linear transformations | Matrix transformations | Linear Algebra | Khan Academy
• Matrix vector products as linear transformations | Linear Algebra | Khan Academy
• Linear transformations as matrix vector products | Linear Algebra | Khan Academy
• Image of a subset under a transformation | Matrix transformations | Linear Algebra | Khan Academy
• im(T): Image of a transformation | Matrix transformations | Linear Algebra | Khan Academy
• Preimage of a set | Matrix transformations | Linear Algebra | Khan Academy
• Preimage and kernel example | Matrix transformations | Linear Algebra | Khan Academy
• Sums and scalar multiples of linear transformations | Linear Algebra | Khan Academy
• More on matrix addition and scalar multiplication | Linear Algebra | Khan Academy
• Linear transformation examples: Scaling and reflections | Linear Algebra | Khan Academy
• Linear transformation examples: Rotations in R2 | Linear Algebra | Khan Academy
• Rotation in R3 around the x-axis | Matrix transformations | Linear Algebra | Khan Academy
• Unit vectors | Matrix transformations | Linear Algebra | Khan Academy
• Introduction to projections | Matrix transformations | Linear Algebra | Khan Academy
• Expressing a projection on to a line as a matrix vector prod | Linear Algebra | Khan Academy
• Compositions of linear transformations 1 | Matrix transformations | Linear Algebra | Khan Academy
• Compositions of linear transformations 2 | Matrix transformations | Linear Algebra | Khan Academy
• Matrix product examples | Matrix transformations | Linear Algebra | Khan Academy
• Matrix product associativity | Matrix transformations | Linear Algebra | Khan Academy
• Distributive property of matrix products | Matrix transformations | Linear Algebra | Khan Academy
• Introduction to the inverse of a function | Matrix transformations | Linear Algebra | Khan Academy
• Proof: Invertibility implies a unique solution to f(x)=y | Linear Algebra | Khan Academy
• Surjective (onto) and injective (one-to-one) functions | Linear Algebra | Khan Academy
• Relating invertibility to being onto and one-to-one | Linear Algebra | Khan Academy
• Determining whether a transformation is onto | Linear Algebra | Khan Academy
• Exploring the solution set of Ax = b | Matrix transformations | Linear Algebra | Khan Academy
• Matrix condition for one-to-one trans | Matrix transformations | Linear Algebra | Khan Academy
• Simplifying conditions for invertibility | Matrix transformations | Linear Algebra | Khan Academy
• Showing that inverses are linear | Matrix transformations | Linear Algebra | Khan Academy
• Deriving a method for determining inverses | Matrix transformations | Linear Algebra | Khan Academy
• Example of finding matrix inverse | Matrix transformations | Linear Algebra | Khan Academy
• Formula for 2x2 inverse | Matrix transformations | Linear Algebra | Khan Academy
• 3 x 3 determinant | Matrix transformations | Linear Algebra | Khan Academy
• n x n determinant | Matrix transformations | Linear Algebra | Khan Academy
• Determinants along other rows/cols | Matrix transformations | Linear Algebra | Khan Academy
• Rule of Sarrus of determinants | Matrix transformations | Linear Algebra | Khan Academy
• Determinant when row multiplied by scalar | Matrix transformations | Linear Algebra | Khan Academy
• (correction) scalar multiplication of row | Matrix transformations | Linear Algebra | Khan Academy
• Determinant when row is added | Matrix transformations | Linear Algebra | Khan Academy
• Duplicate row determinant | Matrix transformations | Linear Algebra | Khan Academy
• Determinant after row operations | Matrix transformations | Linear Algebra | Khan Academy
• Upper triangular determinant | Matrix transformations | Linear Algebra | Khan Academy
• Simpler 4x4 determinant | Matrix transformations | Linear Algebra | Khan Academy
• Determinant and area of a parallelogram | Matrix transformations | Linear Algebra | Khan Academy
• Determinant as scaling factor | Matrix transformations | Linear Algebra | Khan Academy
• Transpose of a matrix | Matrix transformations | Linear Algebra | Khan Academy
• Determinant of transpose | Matrix transformations | Linear Algebra | Khan Academy
• Transpose of a matrix product | Matrix transformations | Linear Algebra | Khan Academy
• Transposes of sums and inverses | Matrix transformations | Linear Algebra | Khan Academy
• Transpose of a vector | Matrix transformations | Linear Algebra | Khan Academy
• Rowspace and left nullspace | Matrix transformations | Linear Algebra | Khan Academy
• Visualizations of left nullspace and rowspace | Linear Algebra | Khan Academy
• Orthogonal complements | Alternate coordinate systems (bases) | Linear Algebra | Khan Academy
• rank(a) = rank(transpose of a) | Matrix transformations | Linear Algebra | Khan Academy