Definitions
Rⁿ
unit vector
line in Rⁿ (parametric form)
hyperplane in Rⁿ  (cartesian or normal form)
linear combination of vectors
dot product of vectors
orthogonal (or perpendicular) vectors
angle between (non-zero) vectors
elementary row operations
echelon and reduced echelon matrices, pivot
rank of a matrix
singular, non-singular matrices
linear transformation
standard matrix of a linear transformation
elementary matrices
inverses and one-sided inverses of matrices
transpose
symmetric matrix

Basic Computations
operations on scalars, vectors and matrices
matrix reduction
solving systems of linear equations
translating problems involving lines, planes, curve fitting, linear combinations and matrix equations to systems of linear equations
passing between cartesian and parametric representatios of lines and planes
projections and angles between vectors
finding matrices of projections, reflections, and rotations in R²
computing inverses of matrices

Conceptual understanding
existence and uniquenes of solutions to Ax=b
good and bad features of vector and matrix operations, especially multiplication
construction of examples and counterexamples