Question

find all scalars c if any exists such that the vector [-3, 3] is a linear combination of the vector [-3,5] and [6,c]]

Answer 1

Linear combination of vector [-3,5] and [6,c] is [-3,3]

a[-3,5] +b [6,c] = [-3,3]

-3a + 6b = -3

a - 2b = 1....(1)

5a + bc = 3...(2)

note that if 1/5 = -2/c, then equation has no solution

so for c =-10, the linear combination is not possible.  

Equation (2) - Equation (1) * 5

b(c+10) = -2

b = -2/(c+10)

(proves again that if c = -10, then equation doesn't have any solution)

so for every real value of c, except -10, vectors can be written as linear combination.