MATH 51 SECTION 1, TUESDAY 1/4/2011
0. Logistics
TA: Henry Adams
(Henry will be back for Thursday’s section. Please email him if you have any logistical questions.)
Email: henrya@math.stanford.edu
Office: 380N
Office hours: For this week only, Thurs 2:15-3:45 and Fri 12:30-2:00
Course website: http://math51.stanford.edu
What is the purpose of section? To review main concepts, ask questions, and do practice problems.
1. Vectors
n
What do the symbols R and R mean?
Symbol R denotes the set of real numbers. We write “x ∈ R” to mean “x is an element of R,” that is, “x
is a real number.”
√
Example. −2.35, 1, 2, and π are real numbers. We write −2.35 ∈ R.
-7,7s
L.A
n
Example. The complex number number 4 + 5i is not in R. We write 4 + 5i ∈
/ R.
Let n be a positive integer. If x1 , . . . , xn ∈ R, then


x1
 x2 
 
 .. 
 . 
xn
is a vector of length n.
 
1
−2.35
Example.
is a vector of length 2, and  π  is a vector of length 3.
3
−3
Symbol Rn denotes the set of vectors of length n.
Example. R2 is the set of vectors in the plane, and R3 is the set of vectors in 3-space.
1
H
There are two basic operations on Rn : addition and scalar multiplication.
−1
2
Problem 1. Let a =
and b =
. What is a + b? How can you draw the three vectors a, b, and a + b
3
1
to visualize this operation?
Problem 2. Let a be as above. What is 2a? What is − 12 a? Draw the three vectors a, 2a, and − 12 a.
2. Linear combinations
Let v1 , . . . , vk be vectors in Rn . A vector v of the form
v = c1 v1 + c2 v2 + · · · + ck vk ,
where c1 , . . . , ck ∈ R, is a linear combination of {v1 , . . . , vk }.
−1
2
Example. Let a =
and b =
. Then
3
1
−7
= 3a − 2b
7
is a linear combination of {a, b}. From Problems 1 and 2, we know that
1
−2
= a + b = 1a + 1b and
= 2a = 2a + 0b
4
6
are also linear combinations of {a, b}.
7
Problem 3. Write
as a linear combination of {a, b}, where a and b are as above.
−14
[Answer: c1 = −5 and c2 = 1]
−1
Problem 4. Write
as a linear combination of {a, b}, where a and b are as above.
−4
[Answer: c1 = −1 and c2 = −1]
3. Spans
The span of a set of vectors is the set of all linear combinations of these vectors. That is, if v1 , . . . , vk ∈ Rn ,
then
span(v1 , . . . , vk ) = {c1 v1 + · · · + ck vk | ci ∈ R for 1 ≤ i ≤ k}.
2
−1
. Then span(a) = {c1 a | c1 ∈ R} consists of all the vectors on the straight line
3
through a and the origin.
−1 −2
Problem 5. What is span
,
?
3
6
−1 [Answer: span
]
3
1
0
Example. Let e1 =
and e2 =
. It turns out that span(e1 , e2 ) = {c1 e1 + c2 e2 | c1 , c2 ∈ R} = R2 .
0
1
That is, the span is as big as it could possibly be.
x
x
x
0
To see this algebraically, consider an arbitrary vector
∈ R2 . Note that
=
+
= xe1 + ye2 .
y
y
0
y
Can you see this geometrically?
 
 
 
1
0
0
Problem 6. Let e1 = 0, e2 = 1, and e3 = 0. What is span(e1 , e2 , e3 ) ?
0
0
1
−1
2
Example. Let a =
and b =
. Then span(a, b) = R2 .
3
1
Can you see this geometrically?
x
To see this algebraically, consider an arbitrary vector
∈ R2 . We want to find c1 , c2 ∈ R such that
y
x
−1
2
= c1
+ c2
. Equating coefficients, we need
y
3
1
Example. Let a =
x = −c1 + 2c2
y = 3c1 + c2 .
Solving this system of equations, we get
2
1
c1 = − x + y
7
7
3
1
c2 = x + y.
7
7
x
−1
2
Note that indeed
= (− 71 x + 27 y)
+ ( 37 x + 17 y)
.
y
3
1
How does this relate to Problems 3 and 4 above?
1 1 Problem 7. What is span
,
?
1
−1
[Answer: R2 ]
     
1
3 1
Problem 8. What is span 1 , −1 , 4 ? Hint: compare with Problem 7.
0
0
0
( 
)
x
[Answer: y  | x, y ∈ R ]
0
     
1
0 1
Problem 9. What is span 1 , −1 , 0 ? Hint: compare with Problem 7.
0
0
1
[Answer: R3 ]
−2 1 Problem 10. What is span
,
?
14
−7
3
−2
[Answer: span
]
14
4. Parametric representations of lines
We saw in §3 that for vector a 6= ~0, the set span a = {ta | t ∈ R} is the straight line through a and
the origin. How do we represent lines which don’t pass through the origin? For any vector x0 , the set
{x0 + ta | t ∈ R} is the line which passes through x0 with direction parallel to v.
( )
2
−1
Problem 11. Draw the line
+t
|t∈R .
−1
3
( )
( )
2
−1
1
−1
+t
| t ∈ R two different lines or are they
Problem 12. Are
+t
| t ∈ R and
−1
3
2
3
two parameterizations of the same line?
[Answer: two parameterizations of the same line]
( )
( )
2
−1
2
2
Problem 13. Are
+t
| t ∈ R and
+t
| t ∈ R two different lines or are they
−1
3
−1
−6
two parameterizations of the same line?
[Answer: two parameterizations of the same line]
 
0
Problem 14. Find a parametric equation for the line which passes through  1  and is parallel to vector
1.5
 
3
 2 .
−1
1
0
Problem 15. Find a parametric equation for the line which passes through
and
.
0
2
4