Math 311 Review 1
Exam office hours: Tuesday, 10:30-12:30.
Graded homework is placed in the bin on my office door
(PSB 318) and may be picked up after 3:00.
Material Lectures: 1-8. Book: pages 1-74, 84-96.
Definitions
A linear combination of variables. A linear equation, a
system linear equations. A solution to a system of
equations. When systems are equivalent. consistent,
inconsistent systems. The three elementary row
operations. Echelon form and reduced echelon form for
systems of equations and matrices. An augmented matrix
for a system of equations. An m[n matrix, the dimensions
of a matrix. A square matrix of order n. The main
diagonal. Scalars. Matrix addition, scalar multiplication,
negation, matrix multiplication, transpose, In the identity
matrix of order n, Omn and On. A0 = I, A1 = A, A2 = AA, A3 =
AAA, ... . A diagonal matrix. An upper triangular, lower
triangular, scalar matrix. A symmetric, skew symmetric
matrix. A submatrix of a given matrix.
A is the inverse of B. A is invertible, singular. Two
matrices are row equivalent. A system of linear equations
is homogeneous. A trivial solution. An elementary matrix.
A vector in the plane or 3-space. Tail, head, PQ¯ ¯¯. When
vectors are equal, a vector’s magnitude and components.
av, u+v, uv.
Theorems
THEOREM. The elementary row operations don’t change the set of
solutions of a system. Hence the new system produced by any of these
operations is equivalent to the original system.
ASSOCIATIVITY. A+(B+C) = (A+B)+C, A(BC) = (AB)C.
COMMUTATIVITY. A+B = B+A, (none for
[
, AB=BC may fail).
DISTRIBUTIVITY. A(B+C) = AB+AC, (B+C)A = BA+CA.
IDENTITY. A+O = O+A = A, AI = IA = A.
INVERSES. A+(-A) = (-A)+A = O.
SCALAR LAWS. rA = Ar, r(sA) = (rs)A, (r+s)A = rA+sA,
r(A+B) = rA+rB, A(rB) = (Ar)B = r(AB).
TRANSPOSE LAWS. (AT)T = A, (A+B)T = AT+BT, (AB)T = BTAT,
(rA)T = rAT.
THEOREM. When two matrices are partitioned into matrices of
submatrices, the product of the matrices is the same as their product
as matrices of submatrices provided the submatrix products are
defined.
THEOREM. An inverse, if it exists, is unique.
THEOREM. If A and B are invertible, then so is AB and
(AB)-1 = B-1A-1.
THEOREM. If A is invertible, so is AT and (AT)-1 = (A-1)T.
THEOREM. Every nonzero matrix can be reduced to a row
equivalent reduced row echelon matrix.
THEOREM. A homogeneous system with more unknowns than
equations has at least one arbitrary variable and hence has a nontrivial
solution.
LEMMA. For any row operation e, (eA_eB) = e(A_B).
LEMMA. Every elementary row operation is invertible.
LEMMA. For any e, E = e(I) is an invertible matrix and
e(A) = EA for any A. And any product of elementary row
operation matrices is also invertible (since the product of
invertible matrices is invertible).
COROLLARY. If applying a sequence e1, e2, e3, ..., en of row
operations to A gives B, i.e., B = en( ... e3(e2(e1A)) ... ), then
B = En(...(E3(E2(E1A)))...) = (En...E3E2E1)A.
THEOREM. If a sequence of row operations converts (A_I)
to (I_B), then B = A-1.
THEOREM. If A reduces to I, then A is a product of
elementary matrices.
LEMMA. For any A and B, if the ith row of A is all 0, then
so is the ith row of AB. Hence a matrix with a row of 0’s
is not invertible.
LEMMA. An n[n matrix in rref is either In (and hence
invertible) or has a row of 0’s (and hence is singular).
THEOREM. For any n[n matrix A, the following are
equivalent: (1) A is invertible. (2) AX = 0n has only the
trivial solution. (3) AX = B has a unique solution for any
column matrix B. (4) A is row equivalent to In. (5) A is a
product of elementary matrices.
COROLLARY. For any n[n matrix A, the following are
equivalent: (1) A is singular. (2) AX = 0n has a nontrivial
solution. (3) A is row equivalent to a matrix with a row of
zeros.
LEMMA. If A is singular, so is AB for any B.
THEOREM. For n[n matrices, if AB = In, then B = A-1.
THEOREM. With row and column operations, every matrix
can be reduced to a matrix consisting of an identity matrix
0 or more all zero rows below it and 0 or more zero
columns on the right.
Main Techniques Be able to
Solve linear systems of equations via row operations on
their augmented matrices.
Be able to reduce a matrix to reduced row echelon form.
Be able to invert matrices using row operations and write
matrices as products of elementary matrices.
Be able to reduce a matrix to an identity matrix plus
possible added 0 rows and columns.
Suggested Exercises. All homework exercises plus the
recommended excersizes.
Page Problem
7: 3, 7, 11, 15, 21.
18: 3, 5, 7, 15, 19, 23, 25.
27: 9, 11, 21, 23.
38: 5, 7, 9, 13.
38: 23, 25, 27, 29, 37, 39.
57: 1b, 3, 7abc, 11, 13, 15, 17.
67: 3abc, 9, 13.
73: 3abcd.
94: 3, 5, 7ab, 9a, 11a, 17.