Math 311-101 — Test I Review — Summer I, 2016

General Information

Calculators. You may use scientific calculators to do numerical calculations — logs, exponentials, and so on. You may not use any calculator that has the capability of doing algebra or calculus, or of storing course material.

Other devices. You may not use cell phones, computers, or any other device capable of storing, sending, or receiving information.

Topics Covered

Systems & matrices

• Linear systems and reduced row echelon form of a matrix
  • Know the row-reduction algorithm as outlined in my Notes on Row Reduction. Be able to put a matrix in reduced row echelon form. Be able to identify the lead variables and free variables. Be able to solve linear systems via row reduction.
  • Special types of systems: homogeneous, overdetermined, underdetermined.
  • Homogeneous systems. Know the connection with solutions to a general system and the corresponding homogeneous system.
  • Be able to solve for currents and voltages in electrical networks.
• 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 + ... +x_n a_n
    where the a_j's are the columns of A.
  • Inverse of a matrix.
    • Be able to be find the inverse of a matrix or show that a matrix is singular via row reducing [A|I].
    • Know that these are equivalent conditions for A to be invertible:
      1. A is nonsingular. (That is, A⁻¹ exists.)
      2. Ax = 0 has only x = 0 as a solution.
      3. A is row equivalent to I.
      4. det(A) ≠ 0.
      5. The columns of A are linearly independent.
    • Know that these are equivalent conditions for A to be singular:
      1. A is singular. (That is, A⁻¹ doesn't exist.)
      2. Ax = 0 has a nontrivial solution x ≠ 0 as a solution.
      3. A is row equivalent to a matrix with 0's in the last row.
      4. det(A) = 0.
      5. The columns of A are linearly dependent
  • Elementary matrices.
    • Know the three types of elementary matrices and how they correspond to row operations. Be able to solve problems similar to those assigned. Be able to write A or A⁻¹ in terms of elementary matrices.

Determinants

• Basic properties. Know the basic properties for determinants. Be able to calculate the determinant of a matrix via its cofactor expansion about a row or a column, or by using row operations, or by some combination of the two methods.
• Determinants of special matrices. The determinant of an upper triangular, lower triangular, or diagonal matrix is the product of the diagonal entries.
• Row reduction of a matrix A and det(A). Know how det(EA) is related to det(A) for an elementary matrix. Know the determinant of the three types of elementary matrices. Be able to read off the determinant of a matrix from the row operations used to reduce it and its row echelon form.
• 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

• Properties and examples.
  • Closure axioms. Addition: If u and v are vectors, then so is u + v. Multiplication by scalars: If c is a scalar and v is a vector, then c·v is a vector.
  • Special vector spaces: Rⁿ, R^m×n (m×n matrices), P_n (polynomials of degree less than n), C[a,b] (continuous functions on [a,b]), C⁽ⁿ⁾[a,b] (n times continuously differentiable functions on [a,b]).

• Subspaces
  • Know the test to determine whether a subset S of a vector space is a subspace: (i) Is 0 in S? (ii) Is S closed under vector addition? (iii) Is S closed under multiplication by a scalar? (Instead of (i) and (iii), test whether, for arbitary scalars α and β, and arbitrary vectors u and v in S, αu + βv is in S.)
  • Know what inear combinations, spans, and spanning sets are. Be able to determine whether or not a set is a spanning set for a vector space. Be able to show that the span of a set of vectors is a subspace.
  • Null space of a matrix A, N(A) = {x in Rⁿ | Ax = 0}.
  • Row space of A. This is the span of the rows of A.
  • Column space of A. This is the span of the columns of A.

• Linear Independence and Linear Dependence
  • Definition and test for LI and LD sets of vectors. To test whether a set S = {v₁, v₂, ..., v_k) is LI or LD, start with the homogeneous equation
    (∗) c₁v₁ + c₂v₂, ..., c_kv_k = 0
    
    • IF the only scalars for which the equation (∗) hold are c₁ = c₂ = ... = c_k = 0, then S is LI.
    • IF there are nonzero scalars for which (∗) holds are, then S is LD.
  • "Coordinate theorem." (Theorem 3.2 in Leon.) Be able to prove this.

• Basis, Dimension and Coordinates
  • Definition of basis. A set S = {v₁, v₂, ..., v_n} is a basis for a vector space V if and only if (i) S is LI and (ii) S spans V. .
  • Definition of dimension. If V has a basis with n>0 vectors in it, then dim(V) = n. If V ={0}, dim(V) = 0. If V has LI arbitrarily large LI sets in it, dim(V) is infinity.
  • Know the standard bases for P_n, Rⁿ, R^m×n
  • Suppose that a vector space V has dim(V) = n and S = {v₁, v₂, ..., v_m}.
    1. m < n
       • S cannot span V.
       • If S is linearly independent and m < n, then vectors may be added to S to make it into a basis.
    2. m > n
       • If m > n, then S is linearly dependent.
       • If S m > n and spans V, then S may be pared down to be a basis for V.
    3. m = n
       • If m = n and S is linearly independent, than S spans V and is a basis.
       • If m = n and S spans V, then S is linearly independent and is a basis.

• Null, Row, and Column Spaces
  • Null space. N(A) = {x in Rⁿ | Ax = 0}.
  • Row space. This is the span of all of the rows of A, Span(r₁, r₂, ..., r_m).
  • Column space. This is the set of all y such that y = Ax; it is the span of the columns of A, Span(a₁, a₂, ..., a_n).
  • Dimensions of subspaces
    • Nullity. nullity(A) := dim(null space(A)).
    • Rank. rank(A) := dim(row space(A)) = dim(column space(A)).
    • The Rank-Nullity Theorem: rank(A) + nullity(A) = # of columns.
  • Bases. Know how to find bases for the subspaces associated with a matrix. See my notes, Methods for finding bases.

Updated: 6/12/2016 (fjn)