Beginner - Expert Linear Algebra, with Practice in Python.

Explore matrix algebra by defining matrices, their indices and real-number entries; perform addition and multiplication of two matrices, and practice transpose and inverse on single matrices.

Learn how to multiply matrices by ensuring inner dimensions match, producing an m by n result, and computing each entry as a sum of products from corresponding row and column.

Multiply a matrix by a constant, subtract matrices using A plus negative B, and perform the element-wise Hadamard operation, with notes on deep learning backpropagation.

this lecture explains the inverse of a square matrix, the identity matrix, and when inverses exist, and outlines properties like (ab)^{-1}=b^{-1}a^{-1} and the path to computing inverses via echelon form.

Apply row reduction to transform a matrix into echelon form and reduced row echelon form using row operations to reveal the leading entry and prepare for finding the inverse.

Transform matrices with elementary row operations to achieve reduced row echelon form, placing leading entries to the left and zeros below pivots, identifying pivot and free columns.

Learn to obtain a matrix inverse via Gauss-Jordan elimination by row reducing A to the identity and applying the same operations to I, demonstrating invertibility with pivot columns.

Discover how to compute a matrix inverse by row-reducing the augmented matrix [A | I] to row-reduced echelon form, applying identical operations to both sides to reveal the inverse.

Apply matrix algebra to solve systems of linear equations, from two-variable lines to three-variable planes, and understand cases with unique, infinite, or no solutions.

Apply Gaussian elimination to solve systems of linear equations by transforming to echelon form, using pivots and zeros to reveal unique or infinite solutions.

Examine linear transformations through rotation of images. Learn that these mappings take input vectors to output vectors, preserving addition and scalar multiplication.

Explore two methods to obtain a transformation matrix for a linear map: derive the matrix from the transformed basis vectors and confirm by multiplying the matrix by any input vector.

Master special transformation matrices that rotate points anticlockwise about a center and map coordinates to images using cosine and sine.

Demonstrates linear transformations and rotation using Python and NumPy, building transformation matrices, applying them to vectors, and visualizing original and transformed points.

Explore vector addition and scalar multiplication, emphasizing same-dimension validity and the zero vector. Examine the dot product as the key vector operation.

Explore the dot product, a scalar-valued vector operation, verify dimension compatibility with a^T b, and illustrate linearity with alpha a dot b. An example yields 14.

Compute a vector's magnitude in n dimensions via the sum of squares and square root, then form unit vectors by dividing by that magnitude.

Compute the distance between two vectors by the magnitude of A minus B, use |A−B|^2, and apply the cosine rule with the angle between A and B.

Learn how to compute the cross product of two vectors in 3D space to obtain a unit normal vector perpendicular to the inputs, using determinant-based formulas and component notation.

Explore the definition of a vector space as a set with defined addition and scalar multiplication, proving identity, inverses, associativity, and commutativity, with examples like R^n and polynomials.

Explore subspaces of a vector space by testing identity, closure under addition, and closure under scalar multiplication, with examples that show non-subspaces and valid subspaces.

Explore how the span of vectors forms a vector space through linear combinations with constants, generating all vectors. See lines, planes, and three-dimensional space and note when a subspace arises.

Learn how the span of vectors v1 through vn forms a subspace of V by containing the zero vector and being closed under addition and scalar multiplication.

Identify linear independence by checking if any vector is a linear combination of the others, and distinguish dependent from independent vectors using pivot columns in echelon form.

Examine the fundamental subspaces, focusing on the null space of a matrix, solving Ax=0, and deriving a basis as a linear combination of vectors.

Explore column space concepts by extracting pivot columns from echelon form, recognizing the span of linearly independent columns, and using examples to identify the column space of a matrix.

Explore the basis of a subspace by verifying linear independence and the ability to span the subspace, using pivot columns, echelon form, and canonical basis concepts.

Explore how dimension and rank relate in vector spaces, including basis, span, column space, and null space, with a detailed example using five vectors in R4.

Explore inner product spaces by verifying positivity, the zero vector condition, linearity, and symmetry with dot products in two and three dimensions.

Explore orthogonal and orthonormal vectors defined by zero inner products and unit norms, forming orthogonal bases and enabling projections, dot products, and the transpose inverse property for orthogonal matrices.

Explore orthogonal projection of a point onto a line spanned by a vector u in R2, derive alpha = (y·u)/(u·u), and compute the projection y_p.

Build a basis from matrix columns, normalize to unit vectors, and obtain an orthonormal basis via QR decomposition; then apply the normal equation to solve X.

Define the determinant of a square matrix using cofactors and minors, and apply cofactor expansion to relate area, volume changes, and inversion.

Learn to compute a matrix inverse from the determinant using cofactors and the transpose, and know when a determinant equals zero to ensure invertibility.

Compute the area of a parallelogram and the volume of a cube using determinants of matrices, selecting focal points, and forming coordinate-based difference vectors.

Define trace as the sum of diagonal elements and explain its invariance under transpose, cyclic permutations, and its relation to the Frobenius norm via the square root of the trace.

this practical session demonstrates determinant and trace computations in Python with numpy, including trace of a matrix and its transpose, the Frobenius norm, and the trace abc equals trace bca.

Examine the concept of number compositions and their factors, showing how 30's divisors reveal even and composite properties, and relate these ideas to the singular value decomposition.

Explore eigen decomposition of a linear transformation, identifying eigenvectors and eigenvalues of a transformation matrix to understand stretch directions and magnitudes.

Learn diagonalization of a matrix using eigenvalues and eigenvectors, form X and D, and compute powers via A^k = X D^k X^{-1} and e^A = X e^D X^{-1}.

Learn to perform cholesky decomposition by expressing a symmetric positive definite matrix as L times L^T, compute the lower triangular factors, and verify A = L L^T.

Explore the singular value decomposition (SVD) of a matrix using U, Sigma, and V^T, including non-square cases and the connection to eigenvectors of E^T E and E E^T.

Explore low rank approximation in linear algebra by selecting leading singular values and reconstructing matrices, and quantify information restoration and the role of the condition number.

Explore fundamental subspaces through the singular value decomposition A = U Σ V^T, linking the column space and null space to U, Σ, and V.

Learn to compute eigenvalues and eigenvectors of square matrices, form the diagonal matrix of eigenvalues, and reconstruct the original matrix via eigen decomposition.

Explore symmetric matrices, their diagonalization via orthogonal eigenvectors, and the relation A = V D V^T. Learn to compute eigenvalues, orthogonalize eigenvectors, and verify diagonalization with examples.

Examine quadratic forms through x^T E x, distinguishing symmetric and non-symmetric matrices, and determine positive definite, negative definite, or indefinite cases using eigenvalues.