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Problems #9 and #13 from pg.32 are graded.
Problem 9: Write a vector equation equivalent to the given system of equations:
{x2+5x3=04x1+6x2−x3=0−x1+3x2−8x3=0.
Solution: Let →a1=[04−1],a2=[163], and →a3=[5−1−8]. Then it is clear that the vector equation
x1→a1+x2→a2+x3→a3=→0
is a vector equation that is equivalent to the given system of equations.
Problem 13: Determine if →b=[3−7−3] is a linear combination of the vectors formed by the columns of A=[1−42035−28−4].
Solution: We will express this question in the form of a vector equation and then solve that vector equation. Write →a1=[10−2],→a2=[−438], and →a3=[25−4]. Now we see that the question of whether →b is a linear combination of the columns of A is simply asking whether or not the vector equation
x1→a1+x2→a2+x3→a3=→b
has a solution or not. We know that all vector equations can reduce to a system of equations, and in this case the system we must solve is
{x1−4x2+2x3=33x2+5x3=−7−2x1+8x2−4x3=−3,
for which we will write an augmented matrix and reduce to reduced echelon form to get
[1−423035−7−28−4−3]∼[102630015300001]
and we see the system of equations has no solution. Therefore there is no way to write the vector →b as a linear combination of →a1,→a2, and →a3.
