MAT 2401 Handout 4.5
Prototype Rn
R2   x, y  | x, y  R , Standard Basis S  1,0 ,  0,1
1. Every element of R2 can be generated by a linear combinations of elements in S .
2. S is linearly independent.
Spanning Set
Basis
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Example 1 Show that S  1, 0, 0  ,  0,1, 0  ,  0, 0,1 is a basis for R 3 .
 S spans R 3 .
First conclusion
 S is linearly
independent.
Second conclusion
Thus, S is a basis for R 3 .
Final conclusion
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.
3. There are more than one basis. For example, S  1,1, 4  , 1, 0,3 , 1, 1, 0  is also a
basis for R 3 .
4. The standard bases are easy to use but may not be the most efficient (§5).
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Example 2 S  1, x, x 2  is the standard basis for P2 .
 S spans P2 .
First conclusion
 S is linearly
independent.
Second conclusion
Thus, S is a basis for P2 .
Final conclusion
Example 3
 1 0   0 1   0 0   0 0  
S  
,
,
,
  is the standard basis for M 2,2 .
  0 0   0 0  1 0   0 1  
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Fact #1
Fact #2 Suppose a vector space V has 2 basis U and W . They must have the same
number of vectors.
Dimension of a Vector Space
Example 4
 
dim R n 
dim  Pn  
dim  M m ,n  
dim  S2  
4