Question

Determine whether the statement below is true or false. Justify the answer. The weights C₁.....Cₚ in a linear combination C₁V₁ + ... + CₚVₚ, cannot all be zero. Choose the correct answer below. A. The statement is false. Setting all the weights equal to zero results in the vector 0 B. The statement is false. Setting all the weights equal to zero does not result in the vector 0. C. The statement is true. Setting all the weights equal to zero results in the vector 0. D. The statement is true. Setting all the weights equal to zero does not result in the vector 0

Answer 1

The statement is false. Setting all the weights equal to zero results in the vector 0.

Step-by-step explanation:

The statement is false. Setting all the weights equal to zero results in the vector 0.

To determine the validity of the statement, let's consider a linear combination of vectors C₁V₁ + ... + CₚVₚ, where C₁, ..., Cₚ are weights and V₁, ..., Vₚ are vectors.

If all the weights C₁, ..., Cₚ are set to zero, then the linear combination becomes 0V₁ + ... + 0Vₚ which simplifies to the zero vector, denoted as 0. The zero vector has no magnitude or direction, and all its components are zero. Therefore, the statement is false as setting all the weights equal to zero results in the vector 0.