Demonstrate that 1 2 1T 1 3 lT 1 1 lT is not a spanning set

Demonstrate that {(1, 2, 1)^T, (1, 3, l)^T, (1, 1, l)T^} is not a spanning set for R^3 by giving a vector not in its span.

Solution

The span of the vectors is the set a(1,2,1) + b(1,3,1) + c(1,1,1) = (a + b + c, 2a + 3b + c, a + b + c),It is not a spanning set because

Note that any vector whose first and third coordinate are not the same will not be in the span of the three vectors.

eg. (23, 24, 25) ,(2,0,3) is not in the span of the vectors.