Master matrices, vector spaces, and linear transformations
← Back to MathematicsLearn to solve systems of linear equations using elimination and matrix methods.
Master fundamental matrix operations and properties.
Calculate determinants and understand their geometric and algebraic significance.
Find matrix inverses and solve matrix equations.
Understand abstract vector spaces and their fundamental properties.
Find eigenvalues and eigenvectors and understand their applications.
Study linear transformations and their matrix representations.
Learn about inner products, orthogonality, and orthonormal bases.
Study symmetric matrices, quadratic forms, and spectral decomposition.
Apply linear algebra concepts to real-world problems and advanced mathematics.
Build the foundation by learning to solve systems of linear equations systematically.
Understand equations of the form a₁x₁ + a₂x₂ + ... + aₙxₙ = b. Learn to identify coefficients and interpret geometric meaning in 2D and 3D.
Work with multiple linear equations simultaneously. Understand when systems have unique solutions, infinitely many solutions, or no solution.
Master the systematic process of using row operations to solve systems. Learn the three elementary row operations and their applications.
Extend Gaussian elimination to obtain reduced row echelon form. Use this method to find complete solutions efficiently.
Recognize row echelon form (REF) and reduced row echelon form (RREF). Understand leading entries, pivot positions, and their significance.
Identify inconsistent systems (no solution) and dependent systems (infinitely many solutions). Interpret results from row operations.
Apply systems of equations to solve problems in business, engineering, and science. Model situations with multiple constraints and unknowns.
Express solutions involving free variables in parametric form. Understand the relationship between free variables and solution sets.
Learn the fundamental operations with matrices and their algebraic properties.
Understand matrix notation A = [aᵢⱼ], dimensions (m×n), entries, rows, columns. Learn standard matrix terminology and conventions.
Perform matrix addition A + B and scalar multiplication kA. Understand when operations are defined and their basic properties.
Master matrix multiplication AB using row-column products. Understand when multiplication is defined and compute products efficiently.
Learn algebraic properties: associativity, distributivity, and when commutativity fails. Understand differences from scalar arithmetic.
Compute the transpose Aᵀ by interchanging rows and columns. Learn properties of transpose and its interaction with other operations.
Identify and work with special matrices: identity matrix I, zero matrix O, diagonal matrices, and their unique properties.
Understand symmetric matrices (A = Aᵀ) and skew-symmetric matrices (A = -Aᵀ). Learn their properties and applications.
Compute matrix powers Aⁿ for square matrices. Use diagonalization and eigenvalues to simplify calculations when possible.
Master determinant calculations and understand their geometric and algebraic significance.
Understand determinants as functions that assign scalars to square matrices. Learn the fundamental properties that define determinants.
Calculate determinants of 2×2 matrices using ad - bc and 3×3 matrices using the rule of Sarrus or cofactor expansion.
Use cofactor expansion along any row or column to calculate determinants of larger matrices. Understand minors and cofactors.
Learn key properties: linearity, effect of row operations, determinant of products, and determinant of transpose.
Understand how elementary row operations affect determinants: row swaps change sign, scaling multiplies determinant.
Calculate determinants of upper and lower triangular matrices as products of diagonal entries. Use this for efficient computation.
Solve systems of linear equations using Cramer's rule when the coefficient matrix is square and invertible.
Understand determinants as signed volumes: 2×2 determinants give signed areas, 3×3 determinants give signed volumes.
Learn to find and work with matrix inverses to solve matrix equations.
Understand that A⁻¹ is the inverse of A if AA⁻¹ = A⁻¹A = I. Learn when inverses exist and their uniqueness.
Learn the Invertible Matrix Theorem: equivalent conditions for invertibility including non-zero determinant and full rank.
Find matrix inverses using the augmented matrix [A|I] and row operations to obtain [I|A⁻¹]. Master this systematic approach.
Calculate inverses using the adjugate matrix: A⁻¹ = (1/det(A)) × adj(A). Understand when this method is practical.
Solve equations of the form AX = B and XA = B using matrix inverses. Understand left and right multiplication by inverses.
Understand elementary matrices as inverses of row operations. Use them to express matrix operations and understand matrix factorizations.
Factor matrices as A = LU where L is lower triangular and U is upper triangular. Use LU factorization for efficient equation solving.
Apply matrix inverses to solve systems of equations, find inverse transformations, and solve practical problems in various fields.
Understand abstract vector spaces and develop the theory of linear independence and bases.
Learn the abstract definition of vector spaces with addition and scalar multiplication operations satisfying eight axioms.
Master the eight axioms: closure, associativity, commutativity, identity elements, inverses, and distributive properties.
Identify subspaces as subsets that are closed under addition and scalar