Question

Which of the following options correctly represents the condition for vectors to be linearly independent?

A) The determinant of the matrix formed by the vectors is zero.
B) The vectors are orthogonal to each other.
C) The sum of the vectors is the zero vector.
D) The vectors cannot form a basis.

Answer 1

The condition for vectors to be linearly independent is that the sum of the vectors is the zero vector.

Step-by-step explanation:

The correct condition for vectors to be linearly independent is option C) The sum of the vectors is the zero vector.

To determine if vectors are linearly independent, we need to check if the linear combination of the vectors equals the zero vector. If the only solution to the equation a1v1 + a2v2 + ... + anvn = 0 is the trivial solution (where all the coefficients ai are zero), then the vectors are linearly independent.

For example, if we have two vectors v1 and v2, and we have the equation a1v1 + a2v2 = 0, if the only solution is a1 = a2 = 0, then the vectors are linearly independent.