Question

Determine if the following statement is true or false: Any four vectors in R³ are linearly dependent.

1) True
2) False

Answer 1

Any set of four vectors in R³ must be linearly dependent because the maximum number of linearly independent vectors in three-dimensional space is three.

Step-by-step explanation:

The statement 'Any four vectors in R³ are linearly dependent' is true. In a three-dimensional space (R³), the maximum number of linearly independent vectors is three since the basis of R³ consists of three vectors. This forms the concept of the dimension of a vector space. Therefore, if you have four vectors in R³, by the definition of linear dependence, at least one of the vectors can be written as a linear combination of the others, making the set of four vectors linearly dependent.