In this bite size post, I would like to consolidate the last blog by solving, in detail, an exercise.


Suppose I have the set of vectors S defined by:

S={2i-4j+3k , 3i-5j , 36i-66j+27k}

Is this a linearly independent set? What is the span of this set?

It clearly is not at first glance obvious if it is or isn't. It might be tempting to say it must be a linearly independent set because the third vector cannot be obtained by multiplying either of the other two vectors by a scalar. But remember, you are also allowed to add vectors together after scaling! So, what we are again essentially asking is can we find a and b such that

(a)(2i-4j+3k)+(b)(3i-5j)=(36i-66j+27k)  [equation 1]
  Note, that in my original definition I might have confused you. We can also include a "c" term multiplying the vector on the right-hand side of the equation, giving

(A)(2i-4j+3k)+(B)(3i-5j)=(C)(36i-66j+27k)
where A and B are in capitals as they would be different from the a and b we have initially.
Note that, however, I can divide by C to eliminate the constant on the right, and I would end up with the expression that I started with, with a=A/C and b=B/C. Diving an arbitrary constant by another arbitrary constant gives yet another arbitrary constant.

So, let's take equation 1 and see if we can solve for a and b. If we end up with a contradiction, i.e., no consistent set of solutions for a and b, we know that the set must be independent.
After solving
2a+3b=36
-4a-5b=-66
3a+0b=27
  we obtain that indeed the solutions are in fact consistent. (Note, there are three equations with 2 variables. Any two is used to find a and b, and the third to check if the set is linearly dependent)
we obtain that a=9 and b=6 solve all three equations, so the set must be dependent, for any vector in S can be expressed as a linear combination of the other two.

Note that just because a set of vectors are linearly dependent does not mean that they have to lie on the same line. All what this term means visually is that at least one vector is in a sense redundant: it could be erased from the given set of vectors and there would be no "drop" in dimensionality:

We could get rid of any of the three vectors above and the two vectors left would still define a plane.

What about the span?

That is easy. If I take any linear combinations of all 3 of the vectors in this set, what is the dimension of space they would "fill up?" We can answer this question purely algebraically:

 

What we are essentially saying is that as the third vector in this set can be expressed as a linear combination of the other two, we really have only two degrees of freedom, two parameters that we can truly vary independently, since it turns out that c3 depends on c1 and c2.

Of course, it is important to note that we can also say that 3i-5j is a linear combination of the other two vectors, and follow the exact same procedure. Any two of the vectors in S would be enough to define a plane.



The End in Mind

What are we doing all of this work for? What we are essentially aiming to do is to get comfortable with transforming between coordinate systems, namely ones that are defined in terms of vectors. First, it is important to come up with specific criteria that allow us to determine if the vectors we have picked span the space we wanted to. If we wanted to describe any point in three-dimensional space in terms of basis vectors that we have chosen, it is no good if the basis vectors only span a plane!

Suppose that we want to go between two coordinate systems. If I define my basis vectors in terms of the regular cartesian coordinate system, how can I describe a point give in the cartesian coordinate system in terms of my basis vectors? How can you translate into your language of i,j,k if I give you a point described in terms of my basis vectors? Is there a quick way to do this?

To answer this we must delve into the world of matrices,  and what they can be thought of representing. Note: when we transform between coordinate systems, we make some special assumptions. I will discuss all of this in the next blog post.