# Math 2700 Key Concepts

• systems of linear equations and writing them in matrix
form

• augmented matrix

• Elementary Row Operations

• Existence and Uniqueness Questions

• Row Reduction, Echelon, and Reduced Echelon form

• Pivot positons

• Vector notation and equations

• Homogeneous and Inhomogeneous equations and systems

• Linear combinations and span

• Linear independence

• Connections between span, linear independence, existence questions, uniqueness
questions, and pivots

• Linear transformations

– Testing if a transformation is linear

– writing a linear transformation as a matrix

– basic geometric examples: rotation, dilation, shear

– one-to-one (also called injective) and connections to linear
independence and pivots

– onto (also called surjective) and connections to span of the columns and
pivots

– invertible = one-to-one and onto (also called bijective)

– kernel

– range or image

• matrix addition, multiplication, and transpose

• How to invert a matrix

– short-cut for 2 × 2

– general procedure for 3 × 3 and larger

• Invertible Matrix Theorem

• null space

• column space

• determinants

– calculating by expanding by cofactors

– calculating by row operations

– Cramer’s Rule

– Connections between determinant of a linear transformation and volume

• Vector spaces and subspaces: definition and how to decide if a set is a vector
space or subspace

• bases and dimension

• rank

• relationship between rank, dimension of null space, dimension of column space,
and number of columns

• change of basis and coordinates

• Eigenvalues and eigenvectors

– characteristic polynomial

– imaginary eigenvalues

– complex eigenvalues

– Using eigenvalues and eigenvectors to help understand a linear
transformation

• Connections between eigenvalues, eigenvectors, and differential equations as
in the Romeo and Juliet examples

• Diagonalization and its relationship to eigenvalues and eigenvectors

• Dot products

– angle between vectors

– length of vectors

– orthogonal

• orthogonal projections

• orthonormal bases and Gram-Schmidt orthogonalization

• least squares problems