Page 110 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes (f) The equations in the system RX = 0 are
x + 2x + 3x = 0
1 2 4
x + 4x = 0
3 4
x = 0
5
Thus V consists of all columns of the form

  2x  3x 4 
2
 x 
 2 
X    4x 4 
 
 x 4 
 0 
 
where x and x are arbitrary.
2 4
(g) The columns

  2   3
 1   0 
   
 0   4
   
 0   1 
 0   0 
   
form a basis of V.
(h) The equation AX = Y has solutions X if and only if

–y + y + y = 0
1 2 3
–3y + y + y – y = 0
1 2 4 5
Self Assessment

3
1. In C , let
 = (1, 0, –i),  = (1 + i, 1 – i, 1),  = (i, i, i)
1 2 3
3
Prove that these vectors form a basis for C . What are the co-ordinates of the vector (a, b, c)
in this basis?
2. Let  = (1, 1, –2, 1),  = (3, 0, 4, –1),  = (–1, 2, 5, 2)
1 2 3
Let

 = (4, –5, 9, –7),  = (3, 1, –4, 4),  = (–1, 1, 0, 1)
4
Which of the vectors , ,  are in the sub-space of R spanned by the  ?
i
6.3 Summary

 In this unit it is shown how elementary row operations help us in understanding the basis
n
of the subspace F .
 The detailed examples show how to go from one basis vector to another by means of an
invertible matrix.

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