Linear Algebra

This topic explores how linear algebra solves real-world problems across science and engineering. Learners will understand how matrices and vector spaces are applied in computer graphics, cryptography, search engines, and machine learning.

Determinants are scalar values computed from square matrices that characterize their invertibility and scaling properties. Learners will understand how to calculate determinants, use Cramer's rule, and analyze volume changes under linear transformations.

Eigenvalues and eigenvectors are scalar factors and corresponding vectors that remain in the same direction during a linear transformation. You will understand how to calculate them and apply them to systems of differential equations, principal component analysis, and stability analysis.

Linear algebra is the study of vectors, vector spaces, and linear transformations represented by matrices. Learners will understand how to solve systems of linear equations, compute eigenvalues, and apply these concepts to data science.

Linear independence describes a set of vectors where no vector can be written as a linear combination of the others. You will learn to determine independence using determinants and row reduction, which is essential for identifying bases in vector spaces.

Linear transformations are functions between vector spaces that preserve vector addition and scalar multiplication. You will understand how to represent these mappings using matrices, find their kernel and image, and analyze their geometric effects on space.

Matrices are rectangular arrays of numbers used to represent linear relationships and transform data. You will understand matrix operations, determinants, and inverses, enabling you to manipulate multi-dimensional data and solve complex systems of equations.

Matrix algebra focuses on the rules and operations governing matrix addition, multiplication, transposition, and inversion. You will understand how to manipulate matrix equations, calculate determinants, and apply these algebraic properties to solve systems of linear equations.

Orthogonality is the generalization of perpendicularity to vector spaces, defined by a zero inner product. You will understand how to construct orthogonal bases using the Gram-Schmidt process and apply orthogonal projections to solve least-squares approximation problems.

Systems of linear equations are collections of linear equations sharing common variables. You will understand how to represent these systems using augmented matrices and solve them systematically using Gaussian elimination, back-substitution, and matrix inversion techniques.

Vector spaces are algebraic structures consisting of vectors that can be added and multiplied by scalars. Learners will understand linear independence, bases, dimension, and linear transformations, enabling them to solve systems of linear equations.