Let S = {(1, 2, 3), (1, 0, -1)}. The value of k for which the vector (2, 1, k) belongs to the linear span of S is: (1) 1 (2) 2 (3) 3 (4) 0
Answer (1)
To find the value of k such that the vector ( 2 , 1 , k ) belongs to the linear span of the set S = {( 1 , 2 , 3 ) , ( 1 , 0 , − 1 )} , we need to determine if it can be expressed as a linear combination of the vectors in S .
The linear span of S is the set of all vectors that can be written in the form:
a ( 1 , 2 , 3 ) + b ( 1 , 0 , − 1 ) = ( 2 , 1 , k )
where a and b are scalars.
Expanding this equation, we have:
a + b = 2
2 a = 1
3 a − b = k
From equation (2) 2 a = 1 , we solve for a :
a = 2 1
Substitute a = 2 1 into equation (1):
2 1 + b = 2
b = 2 − 2 1
b = 2 3
Now substitute a = 2 1 and b = 2 3 into equation (3):
3 ( 2 1 ) − 2 3 = k
2 3 − 2 3 = k
k = 0
Thus, the value of k for which the vector ( 2 , 1 , k ) belongs to the linear span of S is 0 .
So, the correct multiple-choice option is:
(4) 0
