Alternative University

Introduction to Linear Algebra

Vectors

n-tuples

A vector, in the most general sense, is an ordered list. Vectors are often referred to as tuples, or n-tuples, where n equals the number of components in the vector.

The following are examples of vectors that are 3-tuples:

( 1, 2, 3 )
( 7, 8, 9 )

Using variables, those vectors could be denoted:

( a, b, c )

where a = 1 for the first vector, a = 7 for the second vector, etc.

In this example, a, b and c are the components of the vector.


The order of components is important. Changing the order of components produces a different vector that is not equal to the original vector.

For example, the following vectors are equal:

( a, b, c )
( a, b, c )

because the order of components is the same. The following vectors are not equal:

( a, b, c )
( b, a, c )

because the order of components is not the same.


Consider counting “apples and oranges”. We define the following 2-tuple to be the count of apples and oranges:

( Apples, Oranges )

where the integer variable “Apples” denotes the number of apples, and the integer variable “Oranges” denotes the number of oranges.

Say you have a batch of 8 apples and 6 oranges, denoted by the following 2-tuple of (Apples, Oranges) displayed here with square brackets instead of parenthesis and without a comma separating the components:

[  8   6  ]

As mentioned, the order of components is important. In this example, the vector is (Apples, Oranges), with the first component denoting the number of apples, and the second component denoting the number of oranges.

Consider the following 2-tuple of (Apples, Oranges):

[  6   8  ]

In this case, there are 6 apples and 8 oranges, instead of 8 apples and 6 oranges, even though the numbers 8 and 6 are still used, because the position of the numbers are different than the preceding 2-tuple.

This strict ordering of vector components is different than “sets”, in “set theory”, where the order of elements in a set does not matter.

Set theory is a branch of mathematics that defines unordered collections of items, in which each item is called an “element” (instead of a component). For example, the following two sets are equivalent in set theory:

{ a, b, c }
{ b, a, c }

because the order of elements in a set does not matter. In vectors, however, the order of the components matters!


For another real world example, a grid of longitude and latitude lines are used to specify a position on the surface of the Earth:

Figure 1: Longitude and latitude on a globe.

The longitude lines (meridians) run North-South, and the latitude lines (parallels) run East-West.

For a left-handed coordinate system, which is most commonly used in cartograpy, latitude is listed first:

( Latitude, Longitude )

because going from the first component to the second would be a clockwise direction (more on this later). This is referred to as a “Lat/Long” position.

However, using a right-handed coordinate system, a position on the Earth would be:

( Longitude, Latitude )

Therefore, when acquiring longitude and latitude positions, you need to check with the source of the data to find out whether they are listing latitude before longitude, or longitude before latitude. Most data sets will specify “Lat/Long”, indicating that longitude and latitude positions will list latitude first.


Vector Addition

When adding vectors, we add the respective components of each vector, producing a new vector with the same number of components. For example:

  [   A    B    C   ]
+ [   F    G    H   ]
________________________

= [  A+F  B+G  C+H  ]

Addition is only performed within each component. This is referred to as a component-wise operation.

Say that we have two batches of apples and oranges, the first batch with 8 apples and 6 oranges, and the second batch with 5 apples and 4 oranges. Denoting each batch as an (Apples, Oranges) vector and adding the vectors:

   [   8   6   ]
+  [   5   4   ]
______________________

=  [  13   10  ]

gives a total of 13 apples and 10 oranges.


Linear Equations

Returning now to a general form o of the equation of a line:

Ax + By  =  C

consider two lines that intersect at the origin:

Figure 2: Two lines intersecting at the origin.

The equations of those lines in Ax + By = C form are:

𝑥 + 𝑦  
−(½)𝑥 + 𝑦  
=  0
=  0

We can list the coefficients AB and C of each of those two line equations as a vector (A, B, C):

[   1    1   0  ]
[ -1/2   1   0  ]

where A = 1 for the line X+Y=0, and A = −1/2 for the other line, etc.

Adding those two vectors together forms a new vector:

  [   1    1   0  ]
+ [ -1/2   1   0  ]
________________________

= [  1/2   2   0  ]

Using the components of the new vector as the coefficients AB and C of a line equation produces this equation:

(½)x + 2y  =  0

which is a line that also intersects the origin:

Figure 3: The red line is produced by adding the equations of the black lines together.

We say that the new line is a “linear combination” of the other lines. More on that shortly.


Scalar Multiplication

Scalar multiplication of a vector is simply multiplying each component of the vector by a scalar.

For example, given a vector 𝐯 = (i, j, k) and a scalar a, the scalar multiplication a𝐯 is:

a𝐯  
a( i, j, k )
=  ( ai, aj, ak )

Linear Combination

A linear combination is a combination of vector addition and scalar multiplication. In general terms, for a set of n scalars ai and a set of n vectors 𝐯i (where i = 1, 2, 3, … n), a linear combination is:

ai 𝐯i

Sigma notation ( ∑ ) symbolizes summation, this expression expanding as follows:

ai𝐯i = a₁𝐯₁ + a₂𝐯₂ + ⋯ + an𝐯n

For n = 2, the linear combination of the two scalars and the two vectors is:

a₁𝐯₁ + a₂𝐯₂

For n = 3, the linear combination of the three scalars and the three vectors is:

a₁𝐯₁ + a₂𝐯₂ + a₃𝐯₃

and so on.

Note that if each scalar ai is equal to one, the vectors are simply added, as in the example above that created a new line equation from two other line equations. In that case, the following two vectors were added together

𝐯₁  
𝐯₂  
=  ( 1, 1, 0 )
=  ( −½, 1, 0 )

in the linear combination

a₁𝐯₁ + a₂𝐯₂

with a and a both equal to 1.

What if a was equal to −2 instead of 1 ? The term a₁𝐯₁ would still equal 𝐯₁ which is (1, 1, 0), but a₂𝐯₂ would become (1, −2, 0), causing the linear combination to form yet another line through the origin:

  [  1   1  0  ]
+ [  1  -2  0  ]
____________________

= [  2  -1  0  ]

Using the components of the new vector as the coefficients AB and C of a line equation produces this equation:

2xy  =  0

which is a line that also intersects the origin:

Figure 4: The equation for the green line is another linear combination of the equations of the two black lines.

We say that the red and green lines are linear combinations of the two black lines. The red and green lines can themselves be used to generate more linear combinations. All linear combinations of these lines pass through the origin. All lines that pass through the origin are linear combinations of these lines. This is known as a “vector space”.


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2020–Aug–11  21:06  UTC