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Mathematics 220 first test
Thursday, July 24, 2014
please show your work to get full credit for each problem






x1 − 3x3
= 1
= 2
1. For the linear system  2x1 + x2 − 2x3
 −x + 3x + 15x = −1
1
2
3


(a) write down the augmented matrix for this system
(b) row-reduce the augmented matrix to echelon form
(reduced echelon is not necessary)
(c) how many pivots does this augmented matrix have ?
(d) how many free variables does this system of equations have?
(e) write down the solutions of this equation system.
2. Perform each operation, or state that the operation is impossible.

"
T
(a) A if A =
1 1 2
3 5 8
#
1
(d) R if R =
5
2
(
3. For the linear system
(b)
"




1
0
0
1
−4 3
3 4
0
1
1
0





"
a
b
#
3x1 + 2x2 = 5
4x1 + x2 = −5
(e)
#
"
(c) B
1
0
1
−1
2
0
4
2
3
3
0
0
−1
if B =
5
0
0
0
1 2
2 4
#
)
(a) rewrite the system in the format Ax = b
(b) use the product of the inverse matrix A−1 and b to compute the solution x.
(c) solve for x1 and x2 using Cramer’s Rule.
4. Classify each set of vectors as linearly independent or linearly dependent:
(a) {v} if v is a non-zero vector in R3
(b) {u , v , w } if w ∈ Span{u , v }
(c) {u , v , w } if all three vectors are in R2
(d) {u , v , w } if the vector equation x1 u + x2 v + x3 w = 0 has only the trivial solution.
(e) {u , v , w } if the volume of the box
equal to six.
(parallelepiped)
determined by the three vectors is
5. Solve for A:
P AP −1 = B
(supposing A, B, and P are 3 × 3 matrices, and that P is invertible)







1 
1

 1
 



6. Is the vector b =  1  in the set Span  1  ,  1 
 ?


 1
2 
−2
(
7. The linear system
)
Ax1 + Bx2 + Cx3 = 3
Kx1 + M x2 + N x3 = 4






−5
1
x1






has solutions  x2  =  2 + t  6  ,
1
0
x3
where t can be set equal to any real number.
Find all solutions of the related linear systems
(
(a)
Ax1 + Bx2 + Cx3 = 6
Kx1 + M x2 + N x3 = 8
)
(
(b)
Ax1 + Bx2 + Cx3 = 0
Kx1 + M x2 + N x3 = 0
)
.
8. Let T be a linear transformation with domain R2 and codomain R3 . If T (x) = Ax, write
down a possible matrix A.
9. An augmented matrix and its row equivalent reduced echelon form are shown below:


1
2
0 −1 3 b1

−3
−6
1
5 −5 b2 


2
4 −2 −6 −2 b3

reduces to

1 2 0 −1 3 −(2b2 + b3 )/4


 0 0 1 2 4 −(2b2 + 3b3 )/4 
0 0 0 0 0 4b1 + 2b2 + b3
Let A be the 3 × 5 coefficient matrix of the first augmented matrix, and denote its columns
by a1 , a2 , a3 , a4 , and a5 . Denote the last column of the first augmented matrix by b.
Let M be the 3 × 5 coefficient matrix of the row equivalent reduced echelon form matrix,
and denote its columns by m1 , m2 , m3 , m4 , and m5 .
(a) find the complete solution set of Ax = 0.
(b) For which vectors b will the corresponding linear system be consistent ?
(c) Is the unit vector e1 in the set Span{a1 , a2 , a3 , a4 , a5 } ?
(d) Is there a set of three linearly independent columns of A ? If so, write down the three
columns.
(e) Is Span{a1 , a2 , a3 , a4 , a5 } = Span{m1 , m2 , m3 , m4 , m5 } ?
If multiplication by A defines a linear function: T (x) = Ax
(f) state the domain and codomain of T .
(g) describe the range of T .
(h) is the function T one-to-one ?
(i) is the function T onto its codomain?


1 0 0

10. Compute the inverse of the matrix M = 
 3 1 0 
−1 1 2


a b c


11. Suppose that A =  r s t  and that the determinant of A
x y z
Use this information to calculate the values of
(a) det(3A) (b) det AT
(c) det A−1
(d)
(=det(A))
is equal to 4.
x
y
z
4a + r 4b + s 4c + t
a
b
c
12. Suppose T is the function which takes a point in R2 , rotates it 90◦ clockwise around
the origin, and then reflects the resulting point across the vertical line x1 = 0. Find the
standard matrix A so that T (x) = Ax
"
13. Find a 2×2 matrix A for which A
3
4
#
"
=
3
4
#
"
and A
−4
3
#
"
=
4
−3
#
.