Master Matrix Algebra: The Ultimate Linear Algebra Course

Define matrices as square or rectangular arrays of entries with m rows and n columns, using A11 notation, and view them as column or row vectors.

Learn how to add and subtract matrices by performing element-wise operations, using a two-by-two example, and confirm that matrix addition is associative and commutative when matrices share dimensions.

Explore matrix multiplication by pairing a 3 by 2 matrix with a 2 by 3 matrix, verify dimension compatibility, and compute the 3 by 3 result via row-by-column dot products.

Explore the three key ideas of matrix multiplication: it is not commutative, dimensionally constrained, and distributive over addition, with intuition and simple proofs for how these properties shape linear algebra.

Learn how to transpose a matrix by flipping entries and swapping dimensions, so rows become columns, turning an m by n matrix into an n by m matrix with A^T.

Learn to compute determinants of square matrices: use ad minus bc for two by two cases, and cofactor expansion with submatrices and alternating signs for three by three matrices.

Learn how to compute the inverse of a 2x2 matrix by evaluating the determinant, forming the adjugate, and applying the (1/det) adjugate formula, illustrated with a numeric example.

Learn to find the inverse of a 3x3 matrix by building an augmented matrix with the identity and applying row operations until the left becomes the identity, yielding the inverse.

Master matrix algebra: the outer product takes two vectors and yields a matrix, computed as u v^T, illustrated with a 3x3 example from u = [2,0,1], v = [-2,3].

Define the inner product as x^T A y, generalizing the dot product. Ensure symmetry, bilinearity, and positive definiteness of A, assessed via the upper-left determinants.

This lecture demonstrates a two-dimensional inner product, proving symmetry and bilinearity, and shows the matrix is positive definite via upper left determinants to confirm a valid inner product.

Explore how a custom inner product defines vector length: length equals the square root of the inner product of x with itself, using the matrix [[2,-1],[-1,1]] and x = [3,2].

Learn to compute the distance between vectors using the inner product by forming x minus y and taking its length, with a concrete x=(2,4) and y=(3,1) example.

Explore calculating the angle between two vectors using a generalized inner product, comparing the standard dot product and a custom inner product, yielding 18.43 degrees and 45 degrees.

Explore common matrices including zero, identity, diagonal and banded matrices, and learn about upper and lower triangular forms, with a covariance matrix example.

Define singular and non-singular matrices by their determinants; a singular matrix has determinant zero and is not invertible, while a non-singular matrix has determinant not zero and is invertible.

Explore orthogonal matrices and how they rotate vectors while preserving length and angles, with properties like orthonormal rows and columns, inverse equals transpose, and determinants ±1 for applications in rotations.

Construct orthogonal matrices A1 and A2 from theta on the unit circle and use them to rotate a vector 90 degrees clockwise and counterclockwise, while verifying orthogonality and determinant properties.

Shows how an orthogonal matrix rotates a vector by 90 degrees, verifies rows and columns are orthogonal and unit, and preserves vector length (2,1) to (1,-2).

Learn how an orthogonal matrix can permute the rows of a matrix using permutation matrices derived from the identity, and how multiplying P by A reorders the rows of A.

Explore the Gram-Schmidt process, an algorithm that transforms a linearly independent set into an orthonormal set of vectors and defines linear independence and orthonormality.

Demonstrate the Gram-Schmidt process on three-dimensional, linearly independent vectors to produce an orthonormal set, using projections, dot products, and normalization to explain each step.

Explore what a system of linear equations is, illustrate with two equations, and introduce matrix operations, row operations, augmented matrices, reduced row echelon form, and Gaussian elimination as solving methods.

Apply gaussian elimination to solve a three-equation, three-unknown system by building an augmented matrix, zeroing below pivots, achieving upper triangular form, and performing back substitution.

Discover LU decomposition as a faster method to solve Ax equals b by decomposing A into L and U and reusing these factors for any B.

Learn how gaussian elimination row operations form a matrix chain that yields an upper triangular U, enabling back substitution and paving the way for lu decomposition with elementary matrices.

Compute the lu decomposition of A by using the inverses of elementary matrices m1 and m2 to obtain l and u, then use augmented matrix for back substitution.

Verify that the original matrix equals the product of l and u. Show how l comes from elementary matrices and u is the resulting upper triangular factor.

Learn why solving a x = b with lu decomposition is faster by precomputing L and U, storing them, and reusing for multiple b's with forward and back substitution.

Master Cramer's rule to solve a system of linear equations using determinants: d for A, and dx, dy, dz formed by replacing columns with b to find x, y, z.

apply cramer's rule to a concrete three by three system, compute det(A) and the dx, dy, dz determinants, and solve for x, y, z.