Solution. To check whether the vectors are linearly independent, we must answer the following question: if a linear combination of the vectors is the zero vector, is it necessarily true that all the coefficients are zeros? Suppose that x 1 ⃗v 1 + x 2 ⃗v 2 + x 3 ( ⃗v 1 + ⃗v 2 + ⃗v 3 ) = ⃗0 (a linear combination of the vectors is the zero vector). Is it necessarily true that x1 =x2 =x3 =0? We have x1⃗v1 + x2⃗v2 + x3(⃗v1 + ⃗v2 + ⃗v3) = x1⃗v1 + x2⃗v2 + x3⃗v1 + x3⃗v2 + x3⃗v3 =(x1 + x3)⃗v1 + (x2 + x3)⃗v2 + x3⃗v3 = ⃗0. Since ⃗v1, ⃗v2, and ⃗v3 are linearly independent, we must have the coeffi- cients of the linear combination equal to 0, that is, we must have x1 + x3 = 0 x2 + x3 = 0 , x3 = 0 from which it follows that we must have x1 = x2 = x3 = 0. Hence the vectors ⃗v1, ⃗v2, and ⃗v1 + ⃗v2 + ⃗v3 are linearly independent. Answer. The vectors ⃗v1, ⃗v2, and ⃗v1 + ⃗v2 + ⃗v3 are linearly independent.
