Page 212 - Matrix Analysis & Applied Linear Algebra

4.4 Basis and Dimension 207

4.4.3. Determine the dimension of the space spanned by the set

1 1 2 1 3
         
 
 
S =  2   0   8   1   3  .
−1 0 −4 1 0
  ,   ,   ,   ,  
 
 
3 2 8 1 6
4.4.4. Determine the dimensions of each of the following vector spaces:
(a) The space of polynomials having degree n or less.
m×n
(b) The space of m × n matrices.
(c) The space of n × n symmetric matrices.
4.4.5. Consider the following matrix and column vector:
−8
 
 
12205
 1 
A =  24318  and v =  3  .


36155  3 
0
Verify that v ∈ N (A), and then extend {v} to a basis for N (A).

4.4.6. Determine whether or not the set

  
2 1
 
3
B =    1 
,
2 −1
 
is a basis for the space spanned by the set
     
1 5 3
 
4
,
8
2
A =       .
,
3 7 1
 
4.4.7. Construct a 4 × 4 homogeneous system of equations that has no zero
coefficients and three linearly independent solutions.
4.4.8. Let B = {b 1 , b 2 ,..., b n } be a basis for a vector space V. Prove that
each v ∈V can be expressed as a linear combination of the b i ’s
v = α 1 b 1 + α 2 b 2 + ··· + α n b n ,
in only one way—i.e., the coordinates α i are unique.