Suppose {v,w} is a linearly independent set in the vector space V.Then the set {v,v+w} is also linearly independent.A. TrueB. False

To determine whether the set {v, v+w} is linearly independent, we need to consider the linear combination of its elements.

Let's assume that there exist scalars a and b such that av + b(v+w) = 0, where 0 represents the zero vector in V.

Expanding this equation, we get av + bv + bw = 0.

Combining like terms, we have (a+b)v + bw = 0.

Since {v, w} is linearly independent, the only way for this equation to hold is if a+b = 0 and bw = 0.

From the first equation, we can solve for a in terms of b as a = -b.

Substituting this into the second equation, we have (-b)w = 0.

Since {v, w} is linearly independent, w cannot be the zero vector. Therefore, b must be 0 in order for (-b)w = 0.

Substituting b = 0 back into the first equation, we have a = 0.

Thus, the only solution to av + b(v+w) = 0 is a = b = 0.

Therefore, the set {v, v+w} is linearly independent.

Therefore, the statement is True.