MAT 2401 Handout 4.5
Basis
Example 1 Show that S  1, 0, 0  ,  0,1, 0  ,  0, 0,1 is a basis for R 3 .
1. From 4.4 Example 2, we know S spans R 3 .
2.
Remarks
1. S  1, 0, 0  ,  0,1, 0  ,  0, 0,1 is called the standard basis for R 3 .
2. For Rn , the standard basis is defined similarly.
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Example 2 Show that S  1,1, 4  , 1, 0,3 , 1, 1, 0  is a basis for R 3 .
2. From 4.4 Example 7, we know S is linearly independent.
1.
Remarks
1. If the determinant is zero, we have no conclusions. In this case, we need to use GJ
elimination.
2. Some may prefer to start with GJ eliminations which can determine the answer every
time.
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Example 3 S  1, x, x 2  is the standard basis for P2 .
1. From 4.4 Example 3, we know S spans P2 .
2.
 1 0   0 1   0 0   0 0  
Example 4 S  
,
,
,
  is the standard basis for M 2,2 .
  0 0   0 0  1 0   0 1  
Basis and Linear Dependence
Example 5 Let U  1, 2  ,  2,3 ,  3, 4   R 2 . Verify that U is linearly dependent.
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Q: Suppose a vector space V has 2 basis U and W . Can they have different number of
vectors?
A:
Dimension of a Vector Space
Example 6
 
dim R n 
dim  Pn  
dim  M m ,n  
dim  S2  
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MAT 2401 Homework 4.5 Name:________________________________
1. Show that if U  v1 , v2 , v3 , v4 , v5  is linearly independent, then V  v1 , v2 , v3  is also
linearly independent.
Hints
Step 1: Assume the conclusion is not true, that is, V is
linearly dependent. By definition, there are a, b, c  R , Partial solutions are given.
Fill in the missing details.
not all zeros such that
av1  bv2  cv3  0
Step 2: Show that U is linearly dependent.
Now,
 v   v   v   v   v  0
1
2
3
4
5
So, U is linearly dependent
Step 3: Since U is linearly independent, the assumption
in Step 1 must be incorrect. So, V is linearly
independent.
You need to find the
coefficients , not all zeros,
such that the linear
combination of
v1 , v2 , v3 , v4 , v5 is zero.
Remarks The last problem illustrates the following fact.
A nonempty subset of a finite set of linearly independent vectors is linearly independent.
In particular, the following is true.
A sub-collection of elements in a basis is also linearly independent.
For example, U  1, 0, 0, 0  ,  0,1, 0, 0  is a subset of the standard basis for R4 . So,
U is linearly independent.
You are going to use this result in the following problems.
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2. Let Dn ,n be the set of n  n diagonal matrices together with standard matrix addition
and scalar multiplication.
(a) Show that D3,3 is a subspace of M 3,3 .
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(b) Find a basis for D3,3 .
Hints
Let S 
Find a basis S for D3,3 that is also
a subset of the standard basis
for M 3,3 . In that case, because of
the remark after problem 1, you do
not need to show that S is linearly
independent.
All you need to do is to show that
it spans D3,3 . That is, you need to
show that every element in D3,3
can be represented by a linear
combination of the vectors in S.
Since S is a subset of the standard basis for M 3,3 , S
is linearly independent.
Therefore S is a basis for D3,3 .
(c) Find dim  D3,3  .
dim  D3,3  
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