1 suppose x1 x2 x3nbspare linearly independent vectors in a

Question

1. Suppose x₁, x₂, x₃ are linearly independent vectors in a vector space V and let

y₁=x₁+x₂+x₃

y₂=x₁+ax₂

y₃=x₂+bx₃

Find the conditions that must be satisfied by a and b to ensure that the vectors y₁, y₂, y₃ are linearly independent.

2. Let w₁,...,w_j be vectors in a vector V and let v_i ∈span{w₁,...,w_j} for all i=1,2,...,k.

Find an example to show that it is not true that span{v1,...,v_k} = span{w₁,...,w_j}.

Under what conditions would span{v1,...,v_k} = span{w₁,...,w_j}?