Algebra

Several Ideas from Algebra are needed to understand a Vector Space. The list below introduces you to some (not all) of the ideas taught in a High School Algebra class.

Identity Elements

  • a+0=a
  • a*1=a
  • For addition, the identity element is 0.
  • For multiplication, the identity element is 1.

Associativity

  • (a+b)+c = a + (b + c)
  • (a*b)*c = a*(b*c)

Commutativity

  • a+b = b+a
  • a*b = b*a

Distributivity

Distributivity of Multiplication over Addition:

a*(b+c) = (a*b) + (a*c) which is the same as a*(b+c) = a*b + a*c

The above is also worded as “multiplication distributes over addition”. We will show this with an example. Let a=3, b=4 and c=5

3*(4+5) = 3*4 + 3*5

3*(9) = 12 + 15

27 = 27

True

Now, we will show that for the math we use, we cannot say that addition distributes over multiplication. If it did, the law would be a+(b*c) = (a+b) * (a+c):

3+(4*5) = (3+4) * (3+5)

3+(20) = (7) * (8)

23 = 56

False

Proportionality

a = bc

Proportionality is one of those words that you know what that means and it’s difficult to explain it, or kind of difficult at least.

Two things are proportional if the two of them always form the same ratio:

(1,2), (2,4), (3,6), etc.

Below we have generalized the trend:

(x, 2x)

Proportionality is found in Vector spaces because we can multiply a vector by a number to create a new vector that is proportional the old vector.

Appendix A

A few other ideas from Algebra have places where they become important, so we will mention them here.

Variables

We might argue the whole point of Algebra was learning to use symbols (usually letters) to hold numbers. We could then do an equation and have a symbol use different values and we could see what happened. A symbol that we knew was going to change numbers was called a variable.

Constants

In a way, constants are the opposite of variables. If we do work with an equation several times the value of a symbol never changes, then we call that letter a constant.

To make things a little more difficult, someone might say all the symbols are variables, and that a variable is being used as a constant. This attitude works for a lot of equations, like a=bc, but there are some instances of a symbol that never changes. “Pi” will always be 3.14159….