MATH125 Lecture 9: Spanning Sets and Linear Independence

1. Linear Systems of Equations 3. Determine the value(s) of h such that the augmented matrix yields a consistent linear system. (b) [14-11-287. 9h] a) 1 5-4 1 0-3 8 1 0-3 81 4. The matrices 0 2 15 -9and 0 5 5are row equivalent. What is the 01 5-2 0 1 5 -2 row operation that transforms the first matrix to the second? 5. If the statement is always true, circle True. If the statement is sometimes false, circle False. In each case, write a careful and clear justification or counterexample. (a) Two equivalent linear systems can have different solution sets. True False (b) A consistent linear system must have a unique solution True False (c) The matrix 01where each represents a non-zero TrueFalse 0 0 1 real number corresponds to a consistent linear system.

zx1 -4x4 = -10 3x2 + 3x3 = 0 x3 + 4x4 = -1 -3x1 + 2x2 + 3x3 + x4 = 5 [1 h -5 2 -8 6] Two matrices arc row equivalent if they have the same number of rows. Elementary row operations on an augmented matrix never change the solution set of the associated linear system. Two equivalent linear systems can have different solution sets. A consistent system of linear equations has one or more solutions.

For each of the following sets, show that the set is or is not a subspace of the appropriate R". {[x1 x2] | x1 + x2 = 1} {[x1 x2 x3] | x1 + x2 0} {[x1 x2 x3] | x1 - 2x2 = x3, x1 + x2 = 0} Row reduce the following matrices to obtain a row equivalent matrix in row echelon form. Show your steps. Show that each of the following sets is linearly dependent by writing a non-trivial linear combination of the vectors that equals the zero vector. {[1 1 1], [2 2 2]} {[3 -2 4], [1 2 -1], [-6 4 -8]} {[2 1 2], [5 4 5], [1 1 1]} Each of the following matrices is an augmented matrix of a system of linear equations. Determine the values of a, b, c, d for which the systems are consistent. In cases where the system is consistent, determine whether the system has a unique solution.