Math 311 Review 2
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 2:00.
No calculators for this or subsequent exams.
Material Lectures: 9-16. Book: pages 96-151, 156-164.
Definitions
A vector space, closed under addition and scalar
multiplication, subspace. The null space and the nullity
of A.
The span of a set of vectors, span S. S spans V. Vectors
v1, v2, ..., vn are linearly dependent or independent. A
basis for a vector space. The standard basis. The
dimension of a vector space V.
[v]W, the coordinate vector of v w.r.t. W. PW
á
T the
transition matrix from T to W.
The row space (column space) of a matrix and its row
rank (column rank).
Theorems
THEOREM. The 0 is unique. So is -u.
LEMMA (CANCELLATION). uv = uvò implies v = vò.
THEOREM. For any vector space V, any uLV, and scalar c:
D 0u = 0.E c0 = 0.
F If cu = 0 then c = 0 or u = 0.G (-1)u = -u
LEMMA(statement only). Given vectors v1, v2, ..., vn written as
column vectors of Rm, if A = [v1| v2 | ... | vn] is the matrix
whose columns are v1, v2, ..., vn, then v1, v2, ..., vn span
Rm iff the rref of A does not have a row of 0's.
LEMMA(statement only). Given vectors v1, v2, ..., vn written as
column vectors of Rm, if A = [v1| v2 | ... | vn] is the the
matrix whose columns are v1, v2, ..., vn, then v1, v2, ..., vn
are independent iff the rref of A has n nonzero rows.
THEOREM. If v1, v2, ..., vn are a basis for V, then every
vector of V can be written in one and only one way as a
linear combination of v1, v2, ..., vn.
COROLLARY(statement only). Suppose V is a vector space.
D If W is a maximal set of independent vectors (i.e., it
can't be extended to a larger set of independent vectors)
then W is a basis.
E If W is a minimal set of vectors which span V (i.e., no
proper subset of W spans V) then W is a basis.
THEOREM(statement only). Suppose U and W are subsets of a
vector space V and suppose U has u elements and W
has w elements.
(D If U is independent and W spans V, then u w.
E If U and W are both bases, then u = w.
COROLLARY. Suppose W is a subset of a vector space V, W
has w elements and V has dimension n.
D W independent implies w n; w > n implies W is
dependent.
E W spans V î w ! n; w < n î W doesn't span V.
F W independent and w = n implies W is a basis.
G W spans V and w = n implies W is a basis.
LEMMA. D PW
á
T[v]T = [v]W for any vector vLV.
(E PW
á
TPT
á
S = PW
á
S.
(F PT
á
W = (PW
á
T)-1
LEMMA. If V=Rn and U is the standard basis and W and T
are any other bases:
D [v]U = vE PU
á
W = (w1 | w2 | w3 ...)
F PW
á
U = (PU
á
W)-1 G [v]W = PW
á
Uv
H PW
á
T = PW
á
UPU
á
T
THEOREM(statement only). The rows of a matrix's rref matrix
are a basis for the row space. The row rank of a matrix
= the number of nonzero rows of its rref matrix.
THEOREM. The row rank equals the column rank of a
matrix.
Main Techniques Be able to
Identify vector spaces and subspaces.
Determine if a set of vectors span a space, if they are
independent, if they are a basis.
Write a vector as a linear combination of other vectors or
determine that it is not.
Write parametric equations for a given line.
Find bases for given subspaces or for a null space.
Given a set of vectors: Column method - find a subset
which is a basis for the space spanned by the vectors.
Row method - find a simple basis for the spanned
space.
Be able to find transition matrices and given [v]S be able
to find [v]T given bases S and T.
Find O such that AX = OX has a nontrivial solution for a
given A.
Suggested Exercises. All homework exercises plus the
recommended excersizes.
Page Problem
102: 3, 5, 7, 9, 13.
111: 1abcd,3abc,5ab,7abc,9abc,11ab,23abcd,31ab.
122: 1abcd, 3abd, 5, 7, 11abc. 13abc.
137: 1abcd, 3abc, 5bc, 7ab, 9a, 11, 17, 19ac, 21, 29.
147: 3, 5, 13, 15, 17, 19.
161: 1, 3, 7, 9, 13, 19, 23, 25.
169: 1, 5, 19ab, 29ab.
Hw 15 Answers
Page 161
2(2) . 4(2) . 8(1) [3,1,3]. 10(1) t2
3t+2.
3
2
−1
1
−1
3
14 (a)
−9
−8
28
,(d)
2
1
3
,(b)
−2−5−2
−1−6−2
121
.
(e)
−21−2
−10−2
4−17
,(c)
2
1
3
,(f)
−9
−8
28