Show linear dependence or independence. Show all stepsalgebraically.
a. let v1= < x1, x2, ... , xn > and v2 = < y1, y2, ..., yn > be vectors in R^n with v1 not equal to 0. Prove that v1and v2 are linearly dependent if and only if v1 is a non-zeromultiple of v2.
b. Suppose v1, v2, and v3, are linearly independent vectors in avector space V. Show that w1, w2, w3, are linearly independentwhere w1 = v1 + v2 + v3
w2 = v1 - v2 - v3
w3 = 2v1 + v2 - v3
Hint: Assume that c1w1 + c2w2 + c3w3 = 0 and show that c1 = c2 =c3 = 0 by replacing w1, w2, w3 in the above equation with theirexpression in terms of v1, v2, v3, and use the fact that v1, v2 andv3 are linearly independent.
c. Suppose S = { v1, v2, ... , vn } is linearly independent.Prove that any non - empty subset of S is also linearlyindependent. Hint: Assume a subset w1, w2, ... , wk of S islinearly dependent. Show that this implies S is linearly dependentwhich is a contradiction.
for a and c yn and vn , the n are subscripts, and the numbersafter the variables are subscripts, i wasnt sure how to type ithere so v1 is v subscript 1. Thank you! Sorry for theconfusion!
