Math 2270
Quiz 2
4
1. Find all vectors in R that are perpendicular
1
1
1
1
to the three vectors
1
1
2 9
,
3 , 9 .
7
4
Let’s label these three vectors as ~v1 , ~v2 , and ~v3 . Then if ~x is a vector that is perpendicular to ~v1 , ~v2 and ~v3 , we
must have
0 = ~x · ~v1 = x1 + x2 + x3 + x4
0 = ~x · ~v2 = x1 + 2x2 + 3x3 + 4x4 .
0 = ~x · ~v3 = x1 + 9x2 + 9x3 + 7x4
Since the augmented matrix for this system has the last column all zeros, I won’t bother to write the last column
when I row reduce. The row reduction goes as follows:
1 0 0 41
1 1 1 1
1 1 1 1
1 0 −1 −2
1 2 3 4 ∼ 0 1 2 3 ∼ 0 1 2
3 ∼ 0 1 0 − 23 .
0 0 1 94
1 9 9 7
0 8 8 6
0 0 −8 −18
Remembering that there is a column of zeros on the other side of the equals sign and translating this matrix back
into equations, we have that
x1 = − 14 x4
x2 = 32 x4 .
x3 = − 49 x4
In other words, if we let x4 = t, then
1
x1
−4
x2
3
2
x3 = t − 9
4
1
x4
is the solution for any choice of t.
2. In Rm we define
0
..
.
0
~ei =
1 ← ith component.
0
.
..
0
If A is an n × m matrix, what is A~ei ? Use the definition of matrix/vector multiplication to justify your answer.
Let the columns of A be ~v1 , . . . , ~vm . Then by definition
A~ei = 0 · ~v1 + · · · + 1 · ~vi + · · · + 0 · ~vm = ~vi .
Therefore, A~ei is the ith column of A.
1
0
