MATH 129A
Summary for Test 2 (2.3, 2.8, 2.9, 4.1, 5.1 – 5.5)
DEFINITIONS:
Invertible matrix
Invertible transformation
Vector space
Vector subspace
Subspace spanned by a set of vectors
Col(A)
Nul(A)
Spanning set of a (sub)space
Basis of a (sub)space
Dimension of a (sub)space
Rank of a matrix
Determinant (cofactor expansion)
Cramer’s rule
Eigenvector of a matrix
Eigenvalue
Eigenspace
Characteristic polynomial (equation)
Eigenvector basis
Similarity of matrices
Diagonalizable matrix
Real/imaginary part of a complex vector
FALL 2015
THEOREMS:
Invertible matrix theorem
Rank A + dim Nul A = _____
Row operations and determinants
det( AT )
, det( A1 )
, det( AB)
, det( I )
Eigenvectors corresponding to distinct eigenvalues are ________________
Similar matrices have the same ___________________________________
A is diagonalizable iff it has _________ linearly independent eigenvectors
Iff it has ____ eigenvalues counting multiplicity and for each eigenvalue the dim of _________________
equals its ____________
1
A 2 x 2 matrix A with a complex eigenvalue a – ib can be written as A  PCP where
P = ____________ and C = ________________
SKILLS:
Find a basis/dim of Col(A) and Nul(A)
Compute the determinant of a given matrix
Use Cramer’s rule to find the unique solution of a system/matrix equation
Compute the area of a parallelogram/volume of a parallelepiped using determinants
Find eigenvalues and corresponding eigenspaces (basis)
Find matrices P and D for a diagonalizable matrix
Find matrices C and P for a 2 x 2 matrix with complex eigenvalues
PROOFS:
Proofs involving properties of invertible matrices
Prove/disprove that given set is a subspace
Proofs involving properties of determinants
Proofs involving similar matrices