1. For each of the statements below, State whether it is true or false, and Motivate your answer with either a proof, a reference to a definition or theorem, or a counterexample. (a) If S = {s1, s2, s3, s4, s5, s6} is linearly independent, then T = {s1 + s2, s2 + s3, s3 + s4, s4 + s5, s5 + s6, s6 + s1} is also linearly independent. (b) If S = {s1, s2, s3, s4} is linearly independent, then U = {s1 − s2, s2 − s3, s3 − s4, s4 − s1} is also linearly independent. (c) Every subset of a linearly dependent set is also linearly dependent. (d) The column vectors of every 4 × 7 matrix are linearly dependent. (e) If A is a 5×5 upper triangular matrix such that all the diagonal enteries of A are non-zero, then the column vectors of A are linearly independent.
1. For each of the statements below, State whether it is true or false, and Motivate your answer
with either a proof, a reference to a definition or theorem, or a counterexample.
(a) If S = {s1, s2, s3, s4, s5, s6} is linearly independent, then T = {s1 + s2, s2 + s3, s3 + s4, s4 +
s5, s5 + s6, s6 + s1} is also linearly independent.
(b) If S = {s1, s2, s3, s4} is linearly independent, then U = {s1 − s2, s2 − s3, s3 − s4, s4 − s1} is
also linearly independent.
(c) Every subset of a linearly dependent set is also linearly dependent.
(d) The column vectors of every 4 × 7 matrix are linearly dependent.
(e) If A is a 5×5 upper triangular matrix such that all the diagonal enteries of A are non-zero,
then the column vectors of A are linearly independent.
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