2. (75 points) You have a collection of n real k-dimensional vectors. Each vector has a weight attached to it. The vector weights are an input of your algorithm and are not necessarily related to the vector length or any other vector property. Design a greedy algorithm to find the set of vectors that are linearly independent and have maximum total weight, and prove its correctness. (Hint: express the greedy vector to be exchanged asa linear combination of vectors in an optimal solution.) 2. (75 points) You have a collection of n real k-dimensional vectors. Each vector has a weight attached to it. The vector weights are an input of your algorithm and are not necessarily related to the vector length or any other vector property. Design a greedy algorithm to find the set of vectors that are linearly independent and have maximum total weight, and prove its correctness. (Hint: express the greedy vector to be exchanged asa linear combination of vectors in an optimal solution.)