Question

Let x1, ,x2, and x3 be linearly independent vectors in R^(n) and let y1 = x2-x1; y2=x3-x2; y3=x3-x1. Are y1, y2,and y3 linearly independent? rove your answer.

Answer 1

Given x1, x2, and x3 are linearly independent in R^n, their linear combinations, y1, y2, and y3, also form a set of linearly independent vectors because any linear combination of y1, y2, and y3 summing to zero implies that all coefficients must be zero.

Step-by-step explanation:

The question asks if the vectors y1 = x2-x1, y2 = x3-x2, and y3 = x3-x1 are linearly independent given that x1, x2, and x3 are linearly independent vectors in R^n. We can determine the linear independence of y1, y2, and y3 by considering a linear combination of these vectors that equals the zero vector:

a1*y1 + a2*y2 + a3*y3 = 0

Substituting the definitions of y1, y2, and y3, we get:

a1*(x2-x1) + a2*(x3-x2) + a3*(x3-x1) = 0

Expanding and grouping like terms, we obtain:

(-a1-a3)*x1 + (a1-a2)*x2 + (a2+a3)*x3 = 0

Since x1, x2, and x3 are linearly independent, the coefficients of these vectors must all equal zero for this equation to hold. Therefore,

-a1-a3 = 0, a1-a2 = 0, a2+a3 = 0.

Solving these equations, we find that a1 = a2 = a3 = 0. Thus, y1, y2, and y3 are also linearly independent.