Linear Algebra and Geometry

Explore linear algebra and geometry with clear explanations, practical examples, and exercises that build a foundation in matrices, linear systems, vectors, and linear transformations for engineers, scientists, and data professionals.

Explore linear equations, Gaussian elimination, matrices, vector spaces, orthogonality, determinants, eigenvalues, and real-world applications in positive definite matrices and linear transformations.

Discover vectors, matrices, and vector spaces, explore linear transformations, and learn to solve linear systems using only linear operations such as addition and scalar multiplication.

Define vector spaces and their seven properties, including closure under addition and scalar multiplication, and relate these concepts to solving linear systems with examples in R2, R3, and complex spaces.

Show how polynomials with real or complex coefficients form vector spaces, closed under addition and scalar multiplication, and exhibit commutativity, associativity, additive identity and inverse, multiplicative identity, and distributive law.

examine two by two matrices over real or complex fields, showing closure under addition and scalar multiplication, commutativity, associativity, additive and multiplicative identities, additive inverses, and distributivity.

Demonstrate the uniqueness of the additive identity and additive inverses in vector spaces, and show zero times v is zero and negative one times v equals minus v.

Subspaces are subsets of a vector space that contain the zero vector and are closed under addition and scalar multiplication, inheriting vector-space properties. Examples include the zero subspace.

Explore subspaces by examining second-order polynomials and lines through the origin; verify zero vector presence, closure under addition, and closure under scalar multiplication in R2.

Show that the intersection of subspaces W and S is a subspace, includes zero, and is closed under addition and scalar multiplication; define the span as all linear combinations of a set.

Define column space of a matrix as the span of its columns, illustrate with 4 by 3 example, and note vector space properties and role in solvability of linear systems.

Compute the null space of a matrix by solving ax=0 with Gauss-Jordan elimination to find pivot and free variables, then express the null space as the span of special solutions.

Understand linear independence: a set of vectors is independent only when no nontrivial linear combination yields the zero vector; a matrix form and the null space illustrate independence and dependence.

Explore the four vector spaces of a matrix—column space, row space, null space, and left null space—and how pivot, rank, and nullity relate via the transpose.

Explore graphs through incidence matrices that encode node flows, and analyze null space, rank, and left null space to understand loops and the Euler formula for graphs.

Explore orthogonality in spaces through the inner product and see how a zero inner product signals orthogonality. Apply these ideas to solving inconsistent systems and forming orthogonal bases.

Explore orthogonal subspaces where every vector in one is perpendicular to every vector in the other, including the zero vector space, with row space orthogonal to the null space.

Learn how to project a vector u onto v, derive projection vector b and projection matrix P, and recognize that P is symmetric and idempotent with column space span(v).

Discover how to obtain the closest solution to an inconsistent linear system by projecting b onto the column space of A and computing x_hat via the projection matrix.

Find the best fitting line by projecting points onto the line, minimizing error, and solve for x hat and y hat via A^T A x hat = A^T b.

The lecture shows that if A has independent columns, then A^T A is invertible by proving its null space contains only the zero vector, implying its columns are linearly independent.

Study orthonormal vectors, their unit length and orthogonality, and how orthogonal matrices preserve inner products. Derive projections and the least-squares solution using x̂ = Q^T b for square orthogonal matrices.

Apply Gram-Schmidt orthogonalization to convert a vector set into an orthonormal basis, build the QR factorization, and derive the Q and R matrices from the original vectors.

Apply Gram-Schmidt orthogonalization to three independent vectors to construct an orthonormal basis by normalizing the first vector and iteratively removing projections of subsequent vectors, then normalizing.

Explore determinants and inverse matrices by building from four fundamental properties, including identity, row swaps, and row scaling, with geometric interpretation and linearity in rows.

Equal rows yield zero; swapping equal rows leaves the determinant unchanged; subtracting a multiple of a row from another does not change it; a zero row makes the determinant zero.

Explore a two by two determinant in linear algebra and geometry, decomposing the first row into sums and applying properties to show the determinant is zero and well-defined.

Perform Gaussian elimination on this 3x3 matrix to simplify the determinant. Subtracting multiples of one row from another leaves the determinant unchanged and the diagonal product yields the determinant.

Explore properties of determinants, including that the determinant of an upper triangular matrix equals the product of its pivots, singular matrices have zero determinant, and det(AB)=det(A)det(B).

Demonstrates how row operations transform a matrix to upper triangular form, distinguishing singular from non-singular cases and computing determinants, including a negative two.

Learn to compute the determinant of a 3x3 matrix using cofactor expansion from the first row with alternating signs. Recognize the determinant's linearity and how 2x2 determinants arise.

Compute the determinant of a 4x4 matrix using the permutation formula by selecting nonzero entries with distinct column indices across rows, evaluating the contributing permutations and signs, yielding 14.

Learn cofactors, defined by deleting ith row and jth column, with a plus‑minus pattern, to compute determinants and inverses; C22 example and determinant as sum of cofactors times corresponding elements.

This example shows how to compute determinants of tridiagonal matrices of ones using cofactor expansion and alternating signs, deriving determinants for sizes 1 through 4.

Derive a recursive formula for the determinant of n-by-n tridiagonal matrices, from the all-ones case to general entries, using cofactors. Reveal a six-element cycle in the all-ones sequence.

Learn to invert a matrix using the determinant and the cofactor matrix, transpose, and divide by the determinant to obtain the inverse.

Compute the inverse of a tridiagonal matrix using determinant and cofactors, form the cofactor matrix, transpose it, and divide by the determinant, then verify via Gaussian elimination.

Solve ax = b using the inverse; x equals a inverse b follows, with the cofactor transposed over det(a) giving x_j as det(b_j)/det(a), where b_j replaces the jth column by b.

Solve a 4x4 system ax=b by applying determinant-based methods. Use Cramer's rule with column replacements and cofactor expansion to find x, y, z, w (0, 1, 2, -1).

Explore the geometric interpretation of determinants: for a 2x2 matrix, the determinant equals the area of the parallelogram formed by its row vectors, derived from the vectors v and u.

Explore eigenvalues and eigenvectors, learn how a matrix transforms vectors, identify eigenvectors as those that stay in the same direction, and study how to compute and apply these concepts.

Explore eigenvectors and eigenvalues of a projection matrix: vectors in the plane stay fixed with eigenvalue 1, while perpendicular vectors map to 0, and the trace equals their sum.

Learn a systematic method to compute eigenvalues and eigenvectors from the definition by forming the homogeneous equation ax − λI x = 0 and using the determinant zero condition.

Compute eigenvalues of a 3×3 matrix by setting the determinant to zero, factor the polynomial, and obtain eigenvalues 1, 6, and -2, then solve corresponding eigenvectors.

Compute eigenvalues and eigenvectors by subtracting lambda from diagonal and forming the determinant, yielding a second-degree polynomial that may be complex, then verify by multiplying the matrix by the eigenvector.

Subtract lambda from the diagonal and take the determinant to obtain a second-degree equation, revealing repeated eigenvalues and a single eigenvector in this example.

Show that eigenvectors corresponding to distinct eigenvalues are linearly independent, using a contradiction with linear combinations and eigenvalue equations to prove no nontrivial relation exists.

Show that A S = S Λ yields A = S Λ S^{-1}, and that A is diagonalizable with n independent eigenvectors; zero eigenvalues yield zero determinant and no inverse.

Explore how to compute powers of a matrix via diagonalization, showing eigenvectors remain unchanged while eigenvalues are raised to the power k.

Compute the exponential of a matrix A using Taylor series. Diagonalize A when possible and apply the diagonalization S and S inverse to exponentiate the eigenvalues.

Explore symmetric matrices: eigenvalues are real and eigenvectors can be chosen orthonormal, yielding an orthogonal matrix of eigenvectors and the decomposition A = Q Λ Q^T.

Explore how every symmetric matrix can be written as a linear combination of perpendicular projection matrices using its eigenvalues, eigenvectors, and a diagonal eigenvalue matrix.

Compute the eigenvalues of the 2x2 symmetric matrix by solving det(A-λI)=0, yielding λ = ±3. Solve for each eigenvector, obtaining orthonormal vectors and confirming orthogonality via their dot product.

This lecture introduces positive definite matrices, symmetric matrices with positive eigenvalues, positive pivots, and subdeterminants positive, and x^T A x > 0 for all nonzero x.

Explore how positive definite Hessians mark minima and how sums and A^T A preserve positive definiteness for non-singular A.

Explore similar matrices via invertible changes of basis, showing A = S Λ S inverse and Λ = S inverse A S, connecting A to its eigenvalues and eigenvectors.

Learn that similar matrices share eigenvalues, with eigenvectors given by the inverse similarity transform, and that similarity is a reflexive, symmetric, and transitive relation.

Discover how singular value decomposition expresses a as u sigma v, using orthonormal bases for the row and column spaces and a diagonal sigma derived from a^T a.

Examine how linear transformations relate to matrices, preserve addition, and respect scalar multiples, demonstrated by matrix multiplication acting on vectors.

Mathematical matrices represent linear transformations by multiplying vectors and factoring out scalars. The lecture also shows additivity and how a transformation is determined by its action on a basis.

We show that every linear transformation from R^n to R^m is represented by a matrix. We express each domain basis vector in the codomain basis and assemble the coefficient matrix.

Construct the derivative as a linear transformation between polynomial spaces, using bases {1, x, x^2, x^3} and {1, x, x^2}. Apply the derivative matrix to coefficient vectors.

Explore changing basis via linear transformations by constructing a matrix with the new basis vectors as columns and applying the correct composition order to convert coordinates.

Demonstrate change of basis in R2 using v1 and v2, compute w's coordinates in beta, build the basis matrix, and use the 2×2 inverse to transform coordinates.

Examine left and right inverses and the pseudo-inverse. When A has independent columns or rows, inverses exist; using SVD yields A^+ = V sigma^+ U^T as the closest inverse.