Math 2270, Fall 2015
Instructor: Thomas Goller
15 September 2015
Quiz #2 Solutions
(1) Determine by inspection (or otherwise) whether the following vectors are linearly independent:
2 3 2 3 2 3
1
0
1
4 2 5 , 405 , 4 2 5 .
3
1
2
Justify your answer. (3 points)
Possible solution: The vectors are linearly
combination of the first two vectors:
2 3 2
1
4 2 5=4
2
dependent since the third vector is a linear
3 2 3
1
0
2 5 + 405 .
3
1
(2) How many pivot columns must a 999 ⇥ 42 matrix have if its columns are linearly independent? (1 point)
Solution: 42.
Comments: The columns of a matrix A are linearly independent when the equation
A~x = ~0 has only the trivial solution, which happens when there are no free variables.
There are no free variables exactly when every column of A is a pivot column.
Math 2270, Fall 2015
Instructor: Thomas Goller
15 September 2015
02 31
02 3 1

1
0
2
(3) Let T : R3 ! R2 be a linear transformation such that T @405A =
and T @415A =
1
1
02 3 1 1
02 3 1

3
1
1
. Use the fact that T is linear to compute T @405A and T @4 15A . (3 points)
4
3
0
Solution:
02 31
0 2 31
02 3 1


3
1
1
2
6
T @405A = T @3 405A = 3 T @405A = 3
=
.
1
3
3
1
1
02 3 1
02 3 2 3 1
02 3 1
02 3 1



1
1
0
1
0
2
1
1
T @4 1 5 A = T @4 0 5 4 1 5 A = T @4 0 5 A T @4 1 5 A =
=
.
1
4
5
0
1
1
1
1
Comments: While it is possible to solve this problem by guessing a matrix A for T
that has the required properties, the solution shown above is cleaner and highlights the
two properties of what it means for T to be linear. The adjective “linear” is extremely
important in linear algebra!
(4) Bonus problem: Another method for testing linear independence of a set of vectors is
to use those vectors as the rows of a matrix A. The rows of A are linearly dependent
exactly when row reduction on A produces an echelon matrix with a row of all zeroes.
Explain why this method works. (Doing an example may help you see what is going on,
but don’t limit your answer to describing what happened in the example.) (1 bonus
point)
Solution: When you row reduce a matrix, you replace the original rows by linear combinations of those rows that have a simpler form (more zero entries). If you end up
with a row of all zeroes, it means that one of the rows of the original matrix is a linear
combination of the other rows of the matrix. (Since you could have swapped rows during
row reduction, the row that is a linear combination of the other rows could be in any
position.)