Every vector in the set S is a linear combination of two vectors, (1, 1, 0) and (1, 0, 1), and every
vector in the set K is a linear combination of two vectors, (1, 2, 1) and (1, 1, 2). Determine
whether S  K and explain your reason. (98 北科大車輛)
Solution : Let u1  (1, 1, 0), u 2  (1, 0, 1), v1  (1, 2,  1), v 2  (1,  1, 2)
Construct a matrix A with u1 , u 2 , v1 , v 2 as rows
1 0
1 0
1 0
1
1
1
1 0
0  1
0  1 1
1
1

 R12 ( 1); R13 ( 1); R14 ( 1) 
 R23 (1); R24 ( 2) 

1  1
0 0
1 2  1
0
0






2
0 0
1  1 2
0  2
0
The rank of A is 2.  v1 and v 2 are in the vector space of u1 and u 2 .
Now consider av1  bv 2  0  ( a  b, 2a  b,  a  2b)  0  a  0, b  0
This implies v1 and v 2 are linearly independen t . The space can be spanned by v1 and v 2
Hence S  K .
Linear System of Equations 1