Linear Algebra and Geometry 1

Learn how to describe points with coordinates in the plane and in space by using axes, origin, and units, and apply Cartesian coordinates and n-space concepts.

Learn the slope-intercept form y = mx + b to describe lines in the plane, noting horizontal lines have zero slope, while vertical lines fall outside this form.

Learn how lines in the plane and planes in three-space are described by normal equations ax+by+cz+d=0. See how simple rewritings, like y=3 or x=2, become normal form examples.

Explore vectors as direction and magnitude, representing displacement in the plane or space, with coordinates delta x, delta y, and delta z, and note that equal vectors share same displacement.

Understand scalars as real numbers used to scale vectors. A scalar of 2 doubles a vector's length; negative scalars reverse direction.

Learn to add and scale vectors using triangle and parallelogram methods, with delta x and delta y, and explore subtraction, position vectors, preparing you for linear combinations.

Explore linear combinations of vectors, using geometric and algebraic illustrations with u and v, and extend to higher dimensions, unit vectors, and applications to numbers and functions.

Explore matrices as rectangular lists of numbers used to solve linear systems, with elements indexed by row and column, and denoted by A in various shapes like 3x4 or 2x3.

Explore linear transformations and linearity, visualizing how matrices map vectors and the plane, preserve linear combinations, and connect matrix multiplication to geometric images.

Explore matrix vector multiplication as a geometrical view of linear transformations from R2 to R2, using e1 and e2, with columns as images and dot products.

Review the core rules for real-number arithmetic—commutative, associative, and distributive laws—along with neutral elements, zero-product property, cancellation, and additive inverses (opposite numbers) and multiplicative inverses.

Prove the Pythagorean theorem and derive the distance between points in 2d and 3d, using delta x, delta y, delta z, and generalize to RN.

Explore sine and cosine functions, their 2π periodic graphs, unit circle, linking cos α and sin α coordinates to Pythagorean identity cos^2 α + sin^2 α = 1.

Demonstrates how the cosine rule generalizes the Pythagorean theorem for any triangle, linking side lengths a, b, c with the included angle gamma and its acute or obtuse nature.

Explore different views of equations—from solving for x to describing geometric objects like lines, circles, and spheres—using geometrical intuition to understand linear algebra and geometry.

Define the solution set as all values that satisfy left-hand side equals right-hand side, illustrated by one, two, and parameterized solutions, including empty sets and geometric interpretations.

In linear algebra and geometry, explore linear equations and their distinction from non-linear equations, learning linear combinations of unknowns, coefficients, and constants with examples and practice.

Explore systems of linear equations, solving for x and y across two, three, or even four equations, with coefficients, constants, and a matrix-like arrangement of unknowns.

Explore how to solve a system of linear equations, identify inconsistent cases with parallel lines, and distinguish homogeneous systems with trivial and non-trivial solutions, including a one-parameter solution set.

Explore solving a two-by-two linear system via graphical methods, find the intersection of two lines, and verify the solution by plugging in values, in linear algebra and geometry.

Explore two by two systems of linear equations and visualize their solution sets: inconsistent parallel lines, exactly one intersection, or infinite solutions along a single line with a one-parameter solution.

Explore how three by two systems create overdetermined configurations, leading to inconsistent results, a unique solution, or infinitely many solutions through parallel lines, common intersection, and line coincidences.

This lecture analyzes possible solution sets of three by three linear systems, showing no solution, a single point, a line, or a plane.

Explore two by three linear systems; two planes either have no solution, intersect along a line (one-parameter), or coincide (two-parameter), and underdetermined cases occur when equations are fewer than unknowns.

Explore the possible solution sets of m by n systems through geometric intuition about lines and planes, distinguishing inconsistent, one solution, and infinitely many cases for overdetermined and underdetermined systems.

Solve a two by two system algebraically by eliminating a variable with opposite coefficients, then substitute to find x and y.

Learn three elementary operations for solving linear systems: multiply by a nonzero constant, swap equations, and add a multiple of one equation to another, preserving the solution set.

Explore the ideas behind Gauss-Jordan elimination and Gaussian elimination, transforming a system of linear equations with elementary operations into a simplified form, enabling direct solving and back substitution.

Explore Gauss–Jordan elimination on a 2×2 system with a unique solution, using elementary operations to create zeros, achieve coefficient one on the pivots, and read off x and y.

Learn how Gaussian elimination solves a two-by-two system using back substitution, contrasting it with Gauss-Jordan elimination by not requiring zeros above the diagonal.

Demonstrates forming the coefficient and augmented matrices from a two by two system and solving it with Gaussian elimination and Gauss-Jordan elimination using row operations.

Learn to construct augmented matrices for linear systems with correct coefficient matrices and variable ordering, using zeros for absent variables, and apply Gaussian and Gauss-Jordan elimination on two examples.

Learn to convert augmented matrices into corresponding systems of equations by reading coefficients, noting zeros, and choosing variables like x, y, z.

Solve a three-equation system using its augmented matrix with Gaussian and Gauss-Jordan elimination, performing row operations and back substitution to obtain x=17/5, y=-7/5, z=-9/5.

Explore solving a 3x3 system with Gaussian and Gauss-Jordan elimination using an augmented matrix, swapping rows, and back substitution, then verify the solution.

Apply gauss-jordan elimination to a common coefficient matrix with augmented right-hand sides to solve two three-by-three systems simultaneously, and interpret the result as three planes intersecting at a point.

Convert x^2, y^2, z^2 to a, b, c and solve the linear system with Gauss-Jordan elimination to find a=1, b=2, c=4, hence x^2=1, y^2=2, z^2=4 and eight sign choices.

Detect and declare inconsistency using Gaussian or Gauss-Jordan elimination on augmented matrices, recognizing zero left-hand side with a nonzero right-hand side signals no solutions.

Gaussian elimination exposes an inconsistent 3x3 system from the augmented matrix, yielding a zero row with right-hand side -7 and no solution.

Learn how to formulate Gaussian and Gauss-Jordan elimination using row echelon form and reduced row echelon form, augmented matrices, leading ones, rank, and solvability for linear systems, including overdetermined cases.

Learn to read solutions from row echelon and reduced row echelon forms, identify leading and free variables, and express parameterized solutions using Gaussian and Gauss-Jordan elimination.

Learn how to perform Gauss-Jordan elimination on an augmented matrix to obtain a reduced row echelon form, read solutions, and distinguish unique, inconsistent, and infinite solutions via an explicit example.

Construct the augmented matrix for the four by six system and perform Gauss-Jordan elimination to obtain leading ones and zeros. Identify leading and free variables to express the solution.

Apply gauss–jordan elimination to two identical left-hand side systems to obtain reduced row echelon form. One system yields a one-parameter solution, while the other is inconsistent.

Apply Gauss-Jordan elimination to a three-by-four system; after row operations you obtain a last row 0 0 0 0 = -1, so no solutions.

Explore Gauss–Jordan elimination on a variable-parameter system. Analyze cases a=3, a=-3, and a not equal to ±3, yielding infinite, inconsistent, or unique solutions.

Apply Gauss–Jordan elimination to a p-dependent linear system, using augmented matrices and case splits at p = -1 and p = 2 to identify unique, infinite, or no solutions.

Master Gauss–Jordan elimination for systems of linear equations by analyzing augmented matrices, handling a as a parameter, and obtaining a one-parameter solution or unique or inconsistent outcomes with back-substitution.

Master solving systems of linear equations using Gauss-Jordan elimination, and apply them to inverses, line–plane intersections, basis changes, and eigenvectors in linear algebra and geometry.

Solve a, b, c in p(x)=ax^2+bx+c using three points to reconstruct a parabola. Compute a, b, c from p(1)=2, p(-1)=6, p(2)=3, yielding a=1, b=-2, c=3.

Compute the cubic polynomial p(x)=ax^3+bx^2+cx+d that passes through four points by solving the resulting 4x4 system for a, b, c, d; verify p(2)=16 and visualize with GeoGebra.

Learn to compute an integral by partial fraction decomposition of a rational function, factoring the denominator and solving for coefficients to obtain the antiderivative.

Decompose the rational function using partial fractions, set up a system for undetermined coefficients, and solve for a, b, c, d, e to obtain the decomposition.

Balance a chemistry equation by forming an overdetermined linear system and solving via Gaussian elimination, yielding a parametric solution and the need to choose t to obtain integer coefficients.

Apply Ohm's law and Kirchhoff's laws to a three-loop circuit, solving for i1, i2, i3 via a linear system and augmented matrix, yielding i1=6, i2=-5, i3=1.

Introduce matrices as rectangular lists of numbers, explain rows and columns, and define matrices as functions from index sets to R, linking matrix-vector multiplication to linear transformations.

Explore different types of matrices, including zero matrices, identity matrices, symmetric matrices, diagonal matrices, and upper and lower triangular forms, with emphasis on the main diagonal and off-diagonal rules.

Add and subtract matrices by performing element-wise operations on pairs of the same size. Practice with a 2x2 example to compute the sum and the difference.

Explore matrix scaling as scalar multiplication, multiplying every matrix entry by a real number alpha. See how the corresponding linear transformation shrinks the grid lines and parallelograms, reducing area.

Learn matrix scaling by multiplying every entry of a matrix by a real scalar, using alpha A and beta B to show same-size results.

Explore why matrix multiplication equals the composition of linear transformations, and how the product encodes mappings from R^n to R^m via columns that are images of unit vectors.

Align A's columns with B's rows to multiply matrices. Compute each entry as the sum of products of the corresponding row of A and column of B.

Multiply matrices by matching columns and rows via dot products to form the product. See why 2×2 and 2×3 multiply, but 2×3 by 2×2 cannot; columns form linear combinations.

Rewrite systems of linear equations as Ax = b using the coefficient matrix A and unknown vector x. Multiply to obtain a three-by-one result for a three-by-four system, illustrating dimensions.

Explore the transposed matrix: its definition, dimensions swap from m by n to n by m, and how rows become columns and columns become rows, with clear examples.

Define and illustrate the trace of a matrix by summing its main diagonal elements in square matrices, with example calculations and brief practice problems.

Evaluate a series of matrix operations on five matrices with specified sizes to determine definability, and report the resulting dimensions for each expression.

Demonstrates the distributive law for matrix scaling over sums and applies it to compute d+e, d-e, and seven scalar multiples, plus trace and AB, BA products.

Explore the properties of matrix operations, including addition, subtraction, scaling, multiplication, and transposition, and the equality conditions for matrices, comparing to real-number rules and identifying which properties hold.

Explore the commutative and associative properties of matrix addition, the zero matrix as the neutral element for each size, and the additive inverse of each matrix.

Matrix multiplication has a neutral element for square matrices: the identity matrix I_n, and multiplying A by I_n on either side yields A, as shown for small n.

Explore the associative law of matrix multiplication, understand the size conditions for multiplying A, B, and C, and see a hands-on verification using a concrete example.

Explore why matrix multiplication is not commutative by examining AB versus BA using a projection matrix and a linear transformation, illustrating how composition depends on order.

Discover why matrix multiplication is not generally commutative; solve for all 2x2 matrices X that commute with a given 2x2 matrix A, yielding a two-parameter family.

Explore two distributive laws for matrices, showing that (A)(B+C) = AB+AC holds while (A+B)C = AC+BC can fail if dimensions misalign; illustrated with A, B, C.

Demonstrates that matrix multiplication lacks the zero-product property; nonzero matrices can yield a zero matrix, unlike real numbers where ab=0 implies a or b is zero.

Explore why there is no cancellation law for matrix multiplication, illustrated by nonzero A and distinct B, C with AB = AC, and prepare for inverse matrices.

Explore inverse matrices for square matrices, define when A has an inverse, show uniqueness, and distinguish singular from nonsingular cases, with a preview of two by two matrix inverses.

Learn to compute the inverse of a two by two matrix with nonzero determinant by swapping diagonal elements, changing the off-diagonal signs, and scaling by the inverse determinant.

Solve your first matrix equation using inverses, learn why there is no matrix division, multiply by inverses, and verify the solution x = [[1, 8], [-1, -6]].

Explore how to compute powers of square matrices, define zero power as the identity, and see that diagonal matrices raise to powers by multiplying diagonal entries.

Learn the transposed matrix rules, including (A^T)^T = A, (A+B)^T = A^T + B^T, (cA)^T = cA^T, and size conditions for AB and B^T A^T with examples.

Clarifies when to multiply A by B by correcting previous size conditions, showing the only requirement is inner dimension match. Proves (AB)^{-1} = B^{-1} A^{-1} for square invertible matrices.

Demonstrate the inverse of a transposed matrix: for a square nonsingular matrix A, show (A^T)^{-1} = (A^{-1})^T using the (AB)^T rule and inverse uniqueness.

Explore various rules for matrix computations, solve for d using inverses and the identity matrix, and simplify expressions with matrix multiplications, inverses, and powers.

Learn the inverse matrices algorithm using Gauss-Jordan elimination, starting with 2×2 and 3×3 examples and extending to any size via augmented matrices and the identity.

Learn to invert a square matrix by the identity method, solving systems until the left becomes the identity and reading the inverse from the right.

Determine if A is invertible and compute A inverse via gauss-jordan elimination; then solve A x = b with x, y, z as a column vector.

In linear algebra, this lecture uses the Gauss-Jordan method to determine when a parameterized matrix is invertible, showing the matrix is invertible for c not equal to 0 or -1.

Solve a two-by-two matrix equation by factoring with the distributive law, using the identity matrix, and multiplying by the inverse to isolate A.

Solve a 2x2 matrix equation where A is multiplied by a scalar, by treating the scalar as a diagonal matrix, applying the distributive law, and isolating A with an inverse.

Solve a 3x3 matrix equation by isolating x, using distributive law to get x(a+2c), invert a+2c, and compute x as b times (a+2c) inverse, with verification.

Multiply the equation by A^{-1} on the left and by B^{-1} on the right to isolate X, yielding X = I + A B^{-1}. Check invertibility using Gauss-Jordan elimination.

Compute the inverse of a 4x4 matrix via Gauss-Jordan elimination with simultaneous row operations on the matrix and identity, yielding an inverse and exposing fractions.

Explore how gauss-jordan elimination uses elementary row operations on matrices. Show how elementary matrices e1, e2, e3 encode scaling, swapping, and adding rows.

Explore how elementary matrices represent row operations, prove their invertibility, and identify inverses as elementary matrices—for row swaps as self-inverses and for row additions as inverse by subtracting.

Discover the invertible matrix theorem, a central result with 20 equivalent conditions linking determinant, invertibility, and solving linear systems in linear algebra and geometry.

Examine four equivalent statements for square matrices, including invertibility, the trivial solution to ax=0, row equivalence to the identity, and a product of elementary matrices, linking to linear systems.

Explore how a system of linear equations has zero, one, or infinitely many solutions, and how two distinct solutions imply infinitely many, via the coefficient matrix A and rank.

Explore two more statements in invertible matrix theorem for n by n matrices, showing that if A is invertible, Ax = b has one solution for every right hand side.

Solve a 3x3 linear system by computing the inverse of the coefficient matrix A and applying x = A^{-1} b, ensuring a unique solution per the invertible matrix theorem.

Determine consistency by elimination in linear algebra: if the coefficient matrix is invertible, ax = b is always consistent; otherwise, require b3 = b1 + b2.

Determine a 2x2 matrix x in a non-square equation x times a 2x3 matrix equals a 2x3 result, solving for its four entries and verifying consistency by multiplication.

Explore why determinants matter in linear algebra, including their link between nonzero determinants and invertible matrices and unique solutions, and learn diverse computational methods for determinants.

Learn to compute the determinant of a 2-by-2 matrix using ad minus bc and explore the standard notations for determinants, including dt(A) and the matrix form.

Explore the geometric meanings of determinants for 2x2 and 3x3 cases, showing how they compute the area of a parallelogram and the volume of a parallelepiped.

Show determinants measure the signed area of a parallelogram and signed volume of a parallelepiped. Explain that determinants multiply under matrix products, reflecting area scaling under composition of linear transformations.

Define the determinant as a function on square matrices, showing linearity in columns, and relate zero or identical columns to zero, and the identity to one, with geometrical meaning.

Interchanging two columns flips the determinant’s sign. Two exchanges can restore it; this rule underpins determinant behavior in matrices and elementary matrices when permuting columns or rows.

Demonstrates that when one column is a linear combination of the others, the determinant equals zero, with arithmetical and geometrical insight into parallelograms and parallelepipeds.

Demonstrates that adding a multiple of one column to another does not change the determinant, via linearity in columns and the zero determinant of identical columns.

Understand that det(kA) = k^n det(A) for an n by n matrix, showing how scaling by k multiplies the determinant and scales area or volume under the transformation.

Explore elementary column operations and their effects on determinants: scaling a column, swapping columns flips the sign, and adding a multiple of one column to another, with worked examples.

Derive the two-by-two determinant directly from the definition, using linearity in columns and column combinations. Show how ad - bc emerges and why the determinant changes sign with column swap.

Explore how to compute a 3x3 determinant from the definition, using linearity of columns and the identity matrix to motivate efficient methods.

Apply the sarrus method for 3x3 determinants by copying the first two columns and summing diagonal products with plus and minus signs; factor to find zeros such as x=0, ±1.

Explore how determinants behave under transposition and row operations, and connect the permutation-based expansion to signs from disorders; learn the symmetry with the transpose and prepare for alternative methods.

Compute determinants efficiently by expanding along rows or columns using minors and cofactors; apply sarrus for small cases and use row or column operations to simplify larger matrices.

Learn to evaluate determinants by row or column reduction to a triangular matrix, then multiply the diagonal, using zeros creation, cofactor expansion, and Gauss-Jordan style row operations that preserve determinant.

Describe how the determinant of an inverse equals the reciprocal of the determinant for invertible matrices. Explain that det(AB)=det(A)det(B) and that elementary matrices have det 1 or -1.

Compute det((1/5) A · A^T · A^{-1}) using det product rules, det(kA)=k^n det(A) with n=3, det(A^T)=det(A), and det(A^{-1})=1/det(A); row operations yield det(A)=-2/25, giving -2/3125 for the full product.

Explore determinant properties for a 3×3 symbolic matrix: scaling, inverse, transposition, and row order effects; compute results like det(-4A)=320 and det((3A)^{-1})=-1/135.

Explores determinant properties in a 5x5 problem, applying scaling, transpose, and powers to compute det(-A), the reciprocal of det(A), det(A^T), and det(A^4), while illustrating the product rule.

Solve a 4x4 determinant equation by inspecting shared rows and columns, revealing an x factor of degree three, with roots at x = 0, 2, and 6.

Compute the determinant with the Sarrus method to obtain a cubic polynomial, then factorize to reveal its zeros: 0 and ±√2; discuss zeros and roots of polynomials.

Use row operations to factor the determinant; extract x-1 from the two rows, yielding roots x = 1 (double), x = -1, and x = 3/2 via a triangular matrix.

Solve a 4×4 determinant equation equal to zero by creating zeros via row and column operations, then expand to reveal roots x = 1 and x = ±1/√3.

Prove the determinant test for invertibility and apply it to identify x values where det(A)=0, showing invertibility holds when the determinant is nonzero.

Explore Cramer's rule for n by n linear systems, proving an explicit solution formula using determinants, with a 2x2 geometric interpretation and a detailed 3x3 example.

Explore Cramer's rule for solving a three by three system, compute determinants of the coefficient matrix and modified matrices, and confirm results by comparing with a Gaussian elimination method.

Compute the inverse matrix using the explicit adjugate formula, built from minors and cofactors, transposed and scaled by the reciprocal of the determinant.

Demonstrates computing a matrix inverse with the explicit formula and notes the determinant 1 minus six a. Shows invertibility only when a is not 1/6 and compares Jacobi's method.

The lecture shows, by induction, that the n×n determinant equals x^{n-2}(x^2−ab). Solve d=0 to find x=0 with multiplicity n−2, and, when ab≥0, x=±√(ab); otherwise only x=0.

Use induction to prove that the n-by-n determinant d_n equals 3^{n+1} minus 2^{n+1}, with base cases d_1=5, d_2=19, and the recurrence d_{n+1}=5 d_n-6 d_{n-1} from first-row expansion.

Solve a trigonometric determinant using the sine of sum formula, showing the determinant is independent of alpha, beta, gamma, delta, via cosine and sine column operations.

Explore determinants with problem-based learning, proving the Vandermonde determinant, showing its cubic dependence on t, and revealing geometric interpretations via triangles and parallelograms.