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MATH2111 Tutorial 9
Linear Independence
As in Rn , a set of vectors {v1 , v2 , . . . , vk } in a vector space V is linearly independent if
t1 v1 + t2 v2 + · · · + tk vk = 0 implies
t1 = t2 = · · · = tk = 0.
A set of vectors that is not linearly independent is said to be linearly dependent.
1. Let f and g be continuous functions on [a, b], and assume that f (a) = 1 = g(b) and f (b) = 0 = g(a). Show that
{f, g} is linearly independent in C[a, b].
2. Suppose A is an n× n matrix such that Ak−1 6= O but Ak = O for some positive integer k. Show that
I, A, A2 , . . . , Ak−1 is linearly independent in Mn×n .
Prepared by Leung Ho Ming
Homepage: http://ihome.ust.hk/~malhm
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Basis for a Vector Space
A set B of vectors in V is a basis for V if B is linearly independent and span B = V . The dimension of V , denoted by
dim V , is the number of vectors in a basis for V .
3. Suppose S = {A1 , A2 , · · · , Amn } is a basis for Mm×n . If U is an invertible m × m matrix, V is an invertible
n × n matrix, show that T = {U A1 V, U A2 V, . . . , U Amn V } is a basis for Mm×n .
4. The set U =
A:A
1
−1
1
1
=
0
−1
1
A is a subspace of M2×2 . Find a basis for U and the dimension of U .
0
2
Procedures for Finding Basis for Null Spaces, Column Spaces and Row Spaces
Suppose A is an m × n matrix.
After solving Ax = 0, a set of basic solutions is a basis for Nul A.
If A is carried to a row-echelon form R, the nonzero rows of R form a basis for Row A.
The columns of A corresponding to the pivot positions of R form a basis for Col A.
Note that dim(Col A) = dim(Row A) = rank A, and we have the Rank Nullity Theorem saying rank A + dim(Nul A) =
n.



 



1 −1/2 0
2 −1 1
1
0
5/2
−3/2
0

−2 1 1 
0
1

 and rref AT = 0 1 1/2 3/2 
 
5. Let A = 


 4 −2 3. rref A = 0
0
0
0 0 0
0
0
0
0
−6 3 0
(a) Find a basis for Nul A.
(b) Find two bases for Col A.
(c) Find two bases for Row A.
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Coordinates Relative to a Basis
Let B = {b1 , . . . , bk } be an ordered basis for V (where the order in listing is taken into account). Every v ∈ V has a
T
unique representation v = c1 b1 +· · · ck bk , the coordinates of v relative to B is the column vector [v]B = [c1 c2 · · · ck ]
k
in R .
The mapping T : V → Rk defined by T (v) = [v]B is a one-to-one and onto linear transformation, whose inverse is also
linear.
1 1
1 0
0 0
1 0
1 2
,
,
is a basis for M2×2 . Find [v]B if v =
.
6. B =
,
0 0
1 0
1 1
0 1
−1 0
T
7. B = 1 + t, 1 − t, t − t2 is a basis for P2 . Find p(t) if [p(t)]B = [1 2 3] .
     
2 
1
 1
8. Let B = 3 , 2 , 3 be a basis for R3 , then the B-coordinate mapping is a linear transformation from


2
1
2
R3 to R3 . Find the standard matrix of the B-coordinate mapping.
4
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