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.

Question

Hallie Le

Mathematics

3 May, 04:18

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.

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Answers (1)

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Antoine Watkins · Answer 1

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.