Question

If u, v, and w are linearly independent, then which of the following sets is also linearly independent?

1) u, u, v
2) u, v, w
3) u, v, u
4) u, v, w, u

Answer 1

The only linearly independent set from the given options is {u, v, w}, assuming that u, v, and w are initially linearly independent. Any set containing the same vector more than once is not linearly independent.

Step-by-step explanation:

If u, v, and w are linearly independent, no vector in the set can be expressed as a linear combination of the others. This means that for any set to be linearly independent, it cannot contain the same vector more than once, as a vector can always be expressed as a linear combination of itself (simply multiplied by the scalar 1).

The only set that maintains linear independence given that u, v, and w are linearly independent is the set {u, v, w} because it consists of the original vectors which are given to be linearly independent.

Any set that contains the same vector more than once such as {u, u, v}, {u, v, u}, or {u, v, w, u} is not linearly independent because the repeated vectors can be expressed as a linear combination of themselves.