Question

Show linear dependence or independence. Show all steps algebraically.

a. let v1= < x1, x2, ... , xn > and v2 = < y1, y2, ... , yn > be vectors in R^n with v1 not equal to 0. Prove that v1 and v2 are linearly dependent if and only if v1 is a non-zero multiple of v2.

b. Suppose v1, v2, and v3, are linearly independent vectors in a vector space V. Show that w1, w2, w3, are linearly independent where 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 their expression in terms of v1, v2, v3, and use the fact that v1, v2 and v3 are linearly independent.

c. Suppose S = { v1, v2, ... , vn } is linearly independent. Prove that any non - empty subset of S is also linearly independent. Hint: Assume a subset w1, w2, ... , wk of S is linearly dependent. Show that this implies S is linearly dependent which is a contradiction.

for a and c yn and vn , the n are subscripts, and the numbers after the variables are subscripts, i wasnt sure how to type it here so v1 is v subscript 1. Thank you! Sorry for the confusion!

Answer 1

I hope this will help you.thank you