Math2318 Linear Alg
Vectors
Ch1.1
PreRead Ch1.1 to follow along. Fill in the blanks based on your reading.
Vectors describe quantities with ___________________ and _________________.
Example: wind velocity (vector describes ________________ & _________________)
Geometrically: use arrows and directed line segments
Vectors have applications in:
•
______________________
•
_____________________
•
•
______________________
_____________________
Geometry and Algebra of Vectors:
Vectors are __________ _____________ ____________ that describe displacement
from pt A to pg B
⃑⃑⃑⃑⃑ = vector notation for vector from pt A (initial point) to pt B (terminal point)
𝐴𝐵
Tail= ____________________
Head= ___________________
Position vector: ⃑⃑⃑⃑⃑
𝑂𝐴= vector from origin to pt A
Components of a vector
Column vectors
Row vectors
Vector notations: _____, _______
Zero vector = 0
Equal vectors have same ________________ and same ______________.
Geometrically, equal vectors are translations of each other
Standard position of a vector: vector with a tail at the origin
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Math2318 Linear Alg
Vectors
Ch1.1
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Math2318 Linear Alg
Vectors
Ch1.1
Vector Operations:
v = [v1, v2] and w = [w1, w2]
v + w = _______________________
cv = _______________________
vectors in R3
Rn = [v1, v2, v3, v4…..vn] is a vector in n-space (nth dimension)
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Math2318 Linear Alg
Vectors
Ch1.1
Thm 1.1 Algebraic Properties of Vectors in Rn
Linear Combination
Vector v is a linear combination of v1, v2, v3…..vk for scalars c1, c2, c3….ck ∋ (such
that) v=c1v1 + c2v2 + c3v3 + ….ckvk. The scalars are called the coefficients of the
linear combination.
Ex1: Is v a linear combination of v1, v2, and v3?
2
1
2
5
𝑣 = [−2] ⃑⃑⃑⃑
𝑣1 = [ 0 ] ⃑⃑⃑⃑
𝑣2 = [−3] ⃑⃑⃑⃑
𝑣3 = [−4]
−1
−1
1
0
Since v = 3v1 + 2v2 – v3, so yes
2
1
2
5
[−2] = 3 [ 0 ] + 2 [−3] − [−4]
−1
−1
1
0
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Math2318 Linear Alg
Vectors
Ch1.1
Ex2: given u = [3, 1] and v = [ 1, 2]
Write w as a linear combination of u and v with coefficients -1 and 2
-
Used with computer data where 0’s and 1’s represent ____/_____,
_______/_______, ______/_______, or _____/______.
Binary vectors are vectors each of whose ______________________________..
𝑍𝑘𝑛 = notation for vectors of length n in Zk
Modular Arithmetic:
+
0
1
0
0
1
1
0
1
1
·
0
0
0
0
1
0
1
Let 0 = even and 1 = odd
So 1+1 is like odd + odd which is even so 1+1 = 0
{0,1} = Z2= integers in modulo 2
Ex2b: Now you try:
Given v=[1, 0, 1, 1, 0] and w = [0, 1, 1, 1, 1]
Find v + w= ___________________
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Math2318 Linear Alg
Vectors
Ch1.1
Z3
+
0
1
2
0
0
1
1
1
2
2
0
1
2
2
·
0
0
1
2
2
0
1
1
2
0
0
1
2
2
0
1
Ex3: what’s 3548 in Z3?
Find 3548 divided by 3 and use remainders (not fractions or decimals)
So 3548 = ___ (3) + ____
We say that 3548 is congruent to 2 (the remainder), in symbols: 3548 ≡ 2
Ex4: in Z3: 2 + 2 + 1 + 2 =
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Math2318 Linear Alg
Vectors
Ch1.1
Ex5: in 𝑍35 , let v= [2, 2, 0, 1, 2] and w = [1, 2, 2, 2, 1]
Find: v+w
Zm = {0, 1, 2, 3, …..m-1}= integers in modulo m (an m-hour clock). A vector of length
n whose entries are in Zm is called an m-ary vector of length n. Set of all m-ary
𝑛
vectors of length n = 𝑍𝑚
Examples for class practice and discussion:
6. Draw a=[3, 0] and b = [ 3, -2]
7. Given A(3, -2) and B (5, 1)
a) find ⃑⃑⃑⃑⃑
𝐴𝐵 algebraically
b) find ⃑⃑⃑⃑⃑
𝐵𝐴 algebraically
⃑⃑⃑⃑⃑
c) draw 𝐴𝐵
d) draw ⃑⃑⃑⃑⃑
𝐴𝐵 as a vector in standard position
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Math2318 Linear Alg
Vectors
Ch1.1
8)
A, B, C, D, E and F are the vertices of a regular hexagon centered at the origin.
Find the components for each vector from the origin to A, B, C, D, E and F.
9) 𝑢
⃑ =[
3
]
−2
1
𝑣=[ ]
1
⃑ +𝒗
⃑
⃑⃑⃑ = 𝒖
a) 𝒘
⃑ −𝒗
⃑
⃑⃑⃑ = 𝟐𝒖
b) 𝒘
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Math2318 Linear Alg
10) 𝑢
⃑ =[
1
]
−1
Vectors
Ch1.1
2
1
𝑣 = [ ], find 𝑤
⃑⃑ = [ ] as a linear combination of u and v
6
1
11) Given binary vectors u and v, find u + v
a) u = [0, 1] and v = [ 1, 1]
b) u = [1, 1, 0, 1] and v = [ 1, 0, 0, 1]
12) in Z3, find 2(2)(2)
13) in Z4: 3 + 2 + 2 + 3 + 1
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Math2318 Linear Alg
Vectors
Ch1.1
14) in Z5: (2+3)(4+1+3+2)
15) Solve each of the following:
a) in Z5: x + 3 = 2
b) in Z6: 3x = 4
c) in Z5: 2x + 3 = 2
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Math2318 Linear Alg
Vectors
Ch1.1
Recap/Summary/Reminders (for student reflection of section)
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