Question

Determine whether the statement below is true or false. Justify the answer. If x and y are linearly independent, and if z is in Span(x, y), then {x, y, z) is linearly dependent. Choose the correct answer below. O A. The statement is true. Since z is in Span{x, y}, z is a linear combination of x and y. Since z is a li dependent B. The statement is true. Vector z is in Span{x,y} and x and y are linearly independent, so z is a sc (x, y, z) is linearly dependent. C. The statement is false. Vector z is in Span{x, y) and x and y are linearly independent, so z must linearly independent. D. The statement is false. Since z is in Span{x, v}. z cannot be written as a linear combination of x

Answer 1

The statement is true because z, being a member of Span(x, y), implies that it can be expressed as a linear combination of x and y resulting in a linearly dependent set.

Step-by-step explanation:

The statement is true that if x and y are linearly independent, and if z is in Span(x, y), then the set {x, y, z} is linearly dependent. This is because z can be expressed as a linear combination of x and y. Therefore, there exists a non-trivial combination of x, y, and z that equals the zero vector, which is the definition of linear dependence.