Download Review topics:

Review topics:
Vectors, vector addition, scalar multiplication
Linear combinations
Matrix algebra, addition, multiplication, properties
Interpretations of matrix multiplication, e.g.
Each column of AB is a combination of cols of A, generated by corresponding
column of B
Each row of AB is a combination of rows of B, generated by corresponding row
of A
An element of AB is a row of A times a column of B
Subspaces of Rn – what is a subspace, how do we recognize a subspace
Space spanned by a set of vectors – matrix form of such a space:
all vectors Ax, same as column space of A, same as space spanned by columns of A
Space defined by homogeneous conditions: All solutions of Ax=0, same as null space of
A
The dimension of a subspace, basis of a subspace
Row echelon, and reduced row echelon form of a matrix – row operations
Four subspaces: Column space, row space, null space, left null space (null space of AT)
Determining a basis for any of the subspaces using elimination
The rank of a matrix and its connection to the dimensions of the subspaces
The general solution of Ax=b: homogeneous plus particular
Determining linear independence of a set of vectors, finding linear dependencies among
vectors
Inverses – calculating inverses using elimination, using inverses in linear equations,
inverse of a product of square matrices. Singular/nonsingular matrices
Determinants: Basic properties and derived properties
The big formula
Effect of row/column operations on the determinant
Laplace expansion formula using minors
Evaluating a determinant using row operations and expansion
Determinant of a triangular matrix
Determinant of AT
What a determinant determines
Formula for the inverse using a determinant
Cramer’s rule
Singular – not invertible (definition) – detA=0 – columns dependent – rows dependent –
rank(A)<n – Ax=0 has a nonzero solution – A not row equivalent to I
Det(AB)=det(A)det(B) (for square matrices only)