⊗Linear Algebra
Vectors, matrices, linear transformations, vector spaces
Subfields
Vectors
Vector operations, dot product, cross product, projection
Matrices
Matrix operations, determinants, inverse, eigenvalues
Linear Transformations
Rotation, scaling, shearing, change of basis
Vector Spaces
Subspaces, basis, dimension, direct sum
Inner Product Spaces
Inner products, norms, orthogonality, functional analysis connection
Concepts
Vectors
A quantity with both magnitude and direction, represented by coordinates.
Vector Operations
Basic operations on vectors: addition, subtraction, scalar multiplication.
Dot Product
The sum of products of corresponding components, resulting in a scalar. Used to find angles between vectors.
Cross Product
Defined for 3D vectors, produces a vector perpendicular to both input vectors.
Matrices
A rectangular array of numbers, used to represent linear transformations and systems of equations.
Matrix Operations
Operations on matrices: addition, scalar multiplication, matrix multiplication.
Determinant
A scalar value for square matrices, indicating invertibility and volume scaling of linear transformations.
Inverse Matrix
Matrix B such that AB = BA = I, where I is the identity matrix.
Systems of Linear Equations (Matrix Form)
Systems of equations can be expressed and solved in matrix form Ax = b.
Eigenvalues and Eigenvectors
For matrix A, λ is an eigenvalue and v is an eigenvector if Av = λv.
Linear Transformation
A function between vector spaces that preserves addition and scalar multiplication.
Vector Space
A set with vector addition and scalar multiplication satisfying specific axioms.
Basis and Dimension
A basis is a set of linearly independent vectors that span the space. Dimension is the number of basis vectors.