Math 220 Group Solve 1 Solutions
Write all explanations in complete, correct English sentences.
1. For each matrix below, determine whether its columns form a linearly independent set.
Since A has two columns and they are not scalar multiples of each
 4 12 
other, the columns of A are linearly independent, a special case of Th
A   1  3
1.7.
  3 8 
7
0
2

B   4  6 5 
 6 13  3
1 5  3 2 
C  0 4  9 18
0 0 0 0 
Using the row operations, 2R1 + R2, 3R1 + R3, and –R2 + R3 give
2 7 0 
B  0 8 5  which shows that B has three pivot positions, so by
0 0  8
the IMT, the columns of B are linearly independent.
There are more vectors in C than there are entries in each vector, so
the vectors are linearly dependent by Th 1.8 (the “too many vectors”
theorem. .
2. For each matrix in problem 1, determine whether the columns span R3.
The matrix A does not have 3 pivot positions (a pivot position for every row), therefore,
by Th 1.4, the columns of A do not span R3.
The matrix B is a 3 x 3 matrix with 3 pivot positions (by #1 abov3), so by the IMT, its
columns span R3.
The matrix C is a 3 x 4 matrix with a row of zeros, so it does not have a pivot position
in every row. Thus, by Th 1.4, the columns of C do not span R3.
3. Find the standard matrix of the linear transformation T: R2 → R2 that reflects points in the
line x2  x1 and then reflects the result in the horizontal x1 -axis.
0 1 
1 0 
1 0 reflects points in the line x2  x1 and 0 1 reflects points in the horizontal x1 -axis.




1 0  0 1 
 0 1
Order matters. Multiply 
to get 



.
0 1 1 0
 1 0
4. Use the inverse of a matrix to solve the following linear system.
5 x1  6 x2  1
 5  6
1
Let A  
and b    , and note x = A-1b.

 7 x1  8 x2  3
 7 8 
 3
3 
 3   1  5 
 x1    4
 4
Using the calculator to find A1  
,
then


 x   3.5  2.5  3  4
 3.5  2.5
   
 2 
5. a. Complete the following definitions using words if possible.
An indexed set v1 , v2 ,vn  of vectors is said to be linearly dependent if there exist
weights c1 , c2 ,cn not all equal to zero that satisfy c1v1  c2v2    cnvn  0 .
If A in an m x n matrix, and x is a vector in Rn, then Ax is the linear combination of the
columns of A using the corresponding entries of x as weights.
b. Complete the following sentences using the words “always”, “never” or “sometimes”.
It is sometimes possible for six vectors to span R5.
If a linear system has two different solutions, then it always has infinitely many
solutions.
It is never possible to find a matrix that is row equivalent to infinitely many matrices in
reduced row echelon form.