linearly dependent set

Question

Let be a linearly independent set.

Which of the following sets are linearly dependent?

1) {u+v,4u+4v}

2) {6u,6v}

3) {u,0}

4) {u,u+v}

Answer 1

If a set of two vectors is a linearly independent set, then the vectors can’t be multiples of each other. Assume u and v are linearly independent.

1. the second vector 4u+4v=4(u+v), which is a multiple of the first vector. Hence the set is not linearly independent.
2. 6u and 6v are not multiples of each other assuming u and v are not multiples of each other. Hence the set is linearly independent.
3. the zero vector is a multiple of any vector, because any vector times zero is zero, so the set is not linearly independent.
4. Assume u+v is a multiple of u, then there should be a scalar b such that u+v=b*u, this implies (1-b)*u=v, which implies v is a multiple of u, which can’t be true because u and v are linearly independent. So this set is also linearly independent.