Notesale: Turn your study into money

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Rn
We will almost always work in Rn , though the value of n will change a
lot
...
xn ]
...
,

...

xn
On the left is a column vector from Rn , on the right is a row vector
...
In
nearly all cases we will use column vectors
...

This form of vector is easy to work with
...
=

...


...
 


...


...
Next we need the
‘scalar multiplication’, which means you multiply the vector by a one dimensional term like a real number (we will stick to the real numbers, though
there are other options)
...


...


...
Taking x, y, w in Rn , a and b ∈ R and the zero vector 0 in Rn (a vector with
nothing but zeros) we get
1

• x+y=y+x
• x + (y + w) = (x + y) + w
• x+0=x
• x + (−1)x = 0
• (a + b)x = ax + bx
• 1x = x
• a(x + y) = ax + ay
• a(bx) = (ab)x
These are all fairly easy to prove using the vector addition and scalar
multiplications for Rn
...
A linear combination of the vectors v1 , v2 , v3 ,
...
Again, very simple, but we will define several key
concepts and properties using linear combinations
...


 = x1   + x2   + · · · + xn 
...


...


...


...


...


...


To be true, that vector based equation has to match at both coordinates
...
The second is
1 = 3a1 + 2a2

=⇒

1 = 3a1 − 2 − 4a1

=⇒

3 = −a1

leading to a1 = −3 and a2 = 5
...


Dot Products
A dot product between two vectors x and y is written simply as x · y and
calculated like so:

 

x1
y1
 x2   y2 

 

x · y = 
...


...


...

If we take the vector v =

3
4

in R2 we can calculate

v · v = 4 × 4 + 3 × 3 = 16 + 9 = 25
...
It
is, in actual fact, the hypotenuse of a triangle with remaining lengths 4 and
3
...
This is not a coincidence, this is actually
how we calculate the ‘norm’ of vectors, written x
...

√
Norm of x, written x = x · x
which works out nicely since x · x is always non-negative
...

• x·y=y·x
• (ax) · y = a (x · y)
• (x + y) · w = x · w + y · w
• x·0=0
These are fairly easy to confirm using the definition
...
Now for
an attempt to discuss what it means
...
To be completely accurate:
x·y= x

y cos(θ)

where θ is the angle between the two vectors
...


Also, recall that cos(90) = 0, so if the two vectors are at 90 degrees, right
angles to each other (perpendicular), then their dot product is zero
...

1
2
1
2

·

−2
−3

·

−2
1

= −2 + 2 = 0

= −2 − 6 = −8



1
·
2
 

2
=2+2=4
1


1
4
 1   0 
  

 0 · 3 =4+0+0−2=2
1
−2

Orthogonality: if two vectors x and y have dot product x · y = 0 then they
are orthogonal
...
Orthogonality applies in more exotic spaces where the vectors cannot be described as having a direction
...
The ‘Projection’ of x onto y is the component
of x that has the same direction as y
...

There is a simple dot product based way to calculate it:
x·y
P rojy (x) =
y
...

• The dot product is necessary to calculate how close x and y are in
terms of direction
...

• Then we get to the division by y · y
...
On top of the expression y turns up twice, so we have its
size multiplied into the expression twice
...

5

Example: Calculate P rojv1 (v2 ), P rojv2 (v1 ) and P rojv3 (v4 ) using




1
0
 −3 
 1 
2
1



v1 =
, v2 =
, v3 = 
 2  , v4 =  3 
3
−1
2
−3
P rojv1 (v2 ) =

v1 · v2
−1
v1 =
v1 · v1
13

2
3

P rojv2 (v1 ) =

v2 · v1
−1
v2 =
v2 · v2
2

1
−1

=
=

2
− 13
3
− 13

−1
2
1
2


...


Cross Products
These are odd
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Dot products take two vectors in Rn , for any n, and output a real value
...
Dot products are zero if the vectors are perpendicular, cross products
are zero if the vectors are parallel (or in opposite directions)
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
 
 

x1
y1
x2 y3 − x3 y2
 x2  ×  y2  =  −(x1 y3 − x3 y1 ) 
...

Example


 
 
  
1
−2
(2)(0) − (1)(0)
0
 2  ×  1  =  −(1)(0) + (−2)(0)  =  0 
0
0
(1)(1) − (2)(−2)
5
The cross product is clearly orthogonal to the original two vectors
...

−2
−2
(2)(2) − (3)(−1)
7
Use the dot product to check that the result is orthogonal to the originals

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