Math 311-505 - Test I Review
General Information
Test I (Wednesday, October 1) will have 5 to 7 questions, some with
multiple parts. It will cover chapters 1 and 2, and sections 3.1-3.2
in Leon's Linear Algebra (Part A). Please bring an
8½×11 bluebook. Problems will be similar to ones
done for homework. You may use calculators to do arithmetic, although
you will not need them. You may not use any calculator that
has the capability of doing either calculus or linear algebra.
Topics Covered
Systems & matrices
- Linear systems
- Solving systems via row reduction.
- Augmented matrix form. Convert a system to and from augmented
matrix form.
- Row operations and equivalent systems. Be able to define the term
equivalent system. Know the three types of row operations and
that they result in an equivalent system.
- Row echelon form of a matrix and reduced row echelon form. Be
able to use Gauss elimination to put a matrix in row echelon form. Be
able to identify the lead variables and free
variables. Be able to use Gauss-Jordan reduction to put a matrix
in reduced row echelon form. (This form makes the connection
between lead variables and free variables explicit.)
Be able to find all solutions of a linear system by row reducing its
augmented matrix and reading off the solution to the resulting
equivalent system.
- Special types of systems: homogeneous, overdetermined,
underdetermined.
- Homogeneous systems. Know the connection with solutions to a
general system and the corresponding homogeneous system.
- Matrices
- Matrix algebra. Sum, product, scalar multiples, row vectors,
column vectors, transpose, symmetric matrix, identity matrix, zero
matrix, size of a matrix, (i,j) entry, notation. Know the "basic matrix trick"
Ax = x1a1 + x2
a2 + ... +xn an
where the aj's are the columns of A.
- Inverse of a matrix. Know how the inverse is defined. Also, know
the terms invertible, nonsingular, and singular. Be able to be find
the inverse of a matrix or show that a matrix is singular via row
reducing [A|I].
- Application to networks and graphs. Adjacency
matrix A. Interpretation of entries in Ak.
- Elementary matrices
- Three types of elementary matrices and correspondence to row
operations.
- Be able to show that if A and B are invertible, then AB is
invertible and (AB)-1 = B-1A-1
- Definition of row equivalence of matrices.
- Be able to show these are quivalent conditions:
- A is nonsingular.
- Ax = 0 has only x = 0 as a
solution.
- A is row equivalent to I.
- LU factorization. Upper triangular, lower triangular and diagonal
matrices. Be able to factor a matrix into LU form.
Determinants
- Basic properties. Know the basic properties for
determinants. Be able to calculate the determinat of a matrix via its
cofactor expansion about a row or a column.
- Determinants of special matrices. The determinant of an
upper triangular, lower triangular, or diagonal matrix is the product
of the diagonal entries.
- Row and column operations. Be able to use row operations
to find a determinant.
- Row reduciton of a matrix A and det(A). Be able to read
off the determinant of a matrix from the row operations used to reduce
it and its row echelon form.
- Elementary matrices. Know the determinant of the three
types of elementary matrices.
- Inverses. Be able to determine whether an n×n
matrix A is invertible from knowing det A.
- Product rule. det(AB)=det(A)det(B).
Vector spaces
- Basic ideas. Addition, multiplication by scalars, and
being closed under addition and scalar multiplication. Notation for
special spaces: Rn, Rm×n,
Pn, C[a,b], Ck[a,b].
- Subspaces
- Know the definition of a subspace.
- Know the test to determine whether a subset S of a vector space V
is a subspace: (i) Is 0 in S? (ii) Is S closed under + ? (iii)
Is S closed under · ? Be able to use it to determine whether
subsets are subspaces.
- Null space of a matrix, N(A). Be able to find the null space of a
matrix. Know the connection between N(A) and solutions to Ax =
0.
- Span(v1, v2, ...,
vn). Linear combinations. Spanning sets.