Question

If x and y are linearly​ independent, and if ​{x​, y​, z​} is linearly​ dependent, then z is in ​Span{x​, y​}.Choose the correct answer below.True / False.

Answer 1

The correct answer is True, as z must be expressible as a linear combination of the linearly independent vectors x and y, putting z in the Span{x, y}.

Step-by-step explanation:

If x and y are linearly independent, and if the set {x, y, z} is linearly dependent, then it must be the case that z can be expressed as a linear combination of x and y. This is because in a linearly dependent set, at least one of the vectors can be written as a combination of the others. Since x and y are linearly independent, they cannot be written in terms of each other, leaving z to be the vector that depends on x and y. Therefore, z is in the Span{x, y}. The correct answer to the question is True.

Answer 2

True

Step-by-step explanation:

We are given that x and y are linearly independent, if {x,y,z} is linearly independent then z is in span{x,y}.

We have to check that given statement is true or false.

Linearly independent set: A set is called linearly independent if any element is not a linear combination of two or more than two elements of the set.

Dependent set: A set is called linearly dependent when any element of a set is the linear combination of two or more than two elements of the set.

x and y are linearly independent and {x,y,z} is linearly dependent.

It means z is a linear combination of x and y .z is span by x and y.

Therefore, the statement is true.