Download Math 212 Review Problems for Test 1

Math 212 Review Problems for Test 1
May 29, 2003
Dr. John G. Del Greco
Name
Instructions. There are no special instructions for these practice problems. In particular,
they will not be graded.
1. Be sure to know how to deÞne (or state in the case of theorems) each of the following.
(You must give deÞnitions and statements like the ones presented in class.)
• linear equation
• particular solution to a linear equation
• general solution to a linear equation
• m×n linear system of equations
• particular solution a linear system of equations
• general solution to a linear system of equations
• consistent linear system of equations
• augmented matrix of a linear system of equations
• three types of row operations
• row echelon form
• pivot positions
• pivot columns
• Fundamental Theorem of Linear Systems
• reduced row echelon form
• free variable
• vector space Rn
• column vector addition
• scalar multiplication
• zero vector
• linear combination of vectors
• span of a Þnite set of vectors in Rn
• geometric interpretation of span{u} and span{u, v} in Rn
• matrix equation Ax = b
• matrix-vector product Av
• linear transformation
2. Find a linear equation in the variables x and y that has the general solution x = 5 + 2t
and y = t.
3. Consider the following system of equations:
x +
x
2x +
y
+ 2z
+ z
+ 3z
y
=
=
=
a
b
c
Show that for this system to be consistent, the constants a, b and c must satisfy
a + b = c.
4. Obtain the general solution to the following system of equations.
x1
2x1
+
+
3x2
6x2
2x1
+
6x2
− 2x3
− 5x3
5x3
−
+
+
2x4
10x4
8x4
+ 2x5
+ 4x5
+ 4x5
−
+
+
=
0
= −1
=
5
=
6
3x6
15x6
18x6
5. Obtain the general solution to the following system of equations. (A system with all
zeros on the right-hand-side is called a homogeneous system of equations.)
2x1
−x1
x1
+ 2x2
− x2
+ x2
− x3
+ 2x3
− 2x3
x3
− 3x4
+
x4
+
+
−
+
x5
x5
x5
x5
=
=
=
=
0
0
0
0
6. For which values of λ will the following system have no solutions, exactly one solution
and inÞnitely many solutions?
x + 2y
3x − y
4x + y
−
3z
+
5z
+ (λ2 − 14)z
=
=
=
4
2
λ+2
7. Solve the following system of nonlinear equations for the unknown angles α, β, and γ
where 0 ≤ α ≤ 2π, 0 ≤ β ≤ 2π and 0 ≤ γ ≤ π. (Hint: Introduce new variables
u = sin α, v = cos β and w = tan γ and make the system linear.)
sin α
4 sin α
6 sin α
−
cos β
+ 2 cos β
− 3 cos β
+ 3 tan γ
− 2 tan γ
+
tan γ
=
=
=
3
2
9
8. Solve the following system of nonlinear equations. (Hint: Introduce new variables
u = x1 , v = y1 and w = 1z and make the system linear.)
1
x
+
2
y
−
4
z
= 1
2
x
+
3
y
+
8
z
= 0
− x1
+
9
y
+
10
z
= 5
2

 
2
1
9. Prove that the following vectors are in Span{u, v, w} where u = 1, v = −1 and
4
3

3
w = 2.
5


−9
(a) −7 
−15

0

(b) 0
0

7
(c) 8
9
     
0
0 
 2
3





2 , 0 , 1 . (Hint: Show that an arbitrary vector
10. Prove that R = Span


2
3
1

a
b is in the span regardless of the values of a, b and c by showing that the linear
c
system below is always consistent.)
2x
2x
2x
+ 3y
+ z
+ z
11. Let T : R3 −→ R3 be the linear transformation
 
x
4 1



T y = 2 -1
2 2
z
=
=
=
a
b
c
deÞned by
 
x
3


3
y .
0
z

a

Find a condition on the constants a, b, and c so that the vector b lies in the range
c
of T . (Hint: Find a condition on the constants a, b, and c such that linear system
below is consistent.)
4x + y + 3z = a
2x − y + 3z = b
2x + 2y
= c
3