Some Background

In order to get up to speed with the mathematics of concern here, we must cover some jargon. Bear with me, it’s all building for your understanding! (Side note: if you are already familiar with these concepts, feel free to skip ahead. No hard feelings!).


SCALARS

First, let’s discuss a scalar. Scalars are mathematical objects with only a magnitude (for instance: temperature on a thermometer, money, my credit score, anything with a purely numeric value). Mathematical operations involving scalars are quite simple: it looks no different than the addition, subtraction, multiplication, and division that we all know!


 

VECTORS

Next, we come to the vector, only slightly more complex than a scalar. A vector is characterized by a magnitude and a direction.

Vectors are defined by a magnitude (length of the arrow) and direction (with respect to some reference point). Image courtesy of  precal101.weebly.com

 

 

A wonderful example would be simple walking directions (walk five blocks due east). Vectors can exist in any number of dimensions you want (two, three, etc.).

Mathematical operations on vectors can be quite simple. For two vectors pointing in the exact same direction, we just add/subtract their magnitudes. But for ones separated by an angle, we must look to geometry and trigonometry for help.

Visually, we can use the “tip-to-tail” method. We take the tip of one vector, and impose the tail of the other. The vector that results from the addition is shown simply by drawing a line from the tail of the first vector to the tip of the second one. Image courtesy of commons.wikimedia.org.

But what if we don’t have a visual representation of these vectors? Here is where we must draw upon trigonometry. Pythagorean theorem comes quite in handy here. If the vectors originate from the same point, have different magnitudes, and different directions, the path ahead is quite straight forward.

  • If the vectors can be made into a right triangle, do so.
  • With two sides known, we know that A2 + B2 = C2.
  • The resulting vector magnitude is found by taking the square root of side C.
Sides A and B are 11 km, following the math gives their resultant vector. Image courtesy of www.physicsclassroom.com

But what about the direction of the resultant vector? Simple! Geometry tells us that the sine of an angle is equal to the opposite side divided by the hypotenuse side.

We see sine=opposite/hypotenuse. The inverse singe gives us the angle that corresponds to the sine value. Image courtesy of www.physicsclassroom.com.

But what if the vectors cannot be made into a right triangle? It’s a pretty easy to understand process. One must superimpose these vectors onto an x-y coordinate system, and break the vectors down into their x and y components. Each vector has a certain amount of x, and a certain amount of y, we just need to break these down.

A great video of this process is located here:

(PLEASE VIEW “MATH IMAGE RESOURCES” PAGE FOR VIDEO SOURCE)


MATRICES

Next, let’s discuss a matrix. According to Wolfram Math World, a matrix is “a concise and useful way of representing and working with linear transformations” (check their page, it’s great stuff!). In my experiences, a matrix is a very handy way for dealing with multiple variables at one time.

Image courtesy of http://www.mathwarehouse.com/algebra/matrix/
Image courtesy of http://www.mathwarehouse.com/algebra/matrix/

How would one go about adding two matrices? For starters, the two matrices must be of the same dimensions, having the same number of columns and rows. A better way to speak of this is saying a matrix has n×m elements. A matrix whose n and m elements are equal is said to be a square matrix (which we will see lots of coming up). Next, start with the upper left corner of one matrix, and if that number to the corresponding one in the next matrix. The answer goes into a new matrix of the same dimensions.

The two numbers in the same position are added, then put into the corresponding position in the new matrix. The same rules apply to subtraction. The matrices here have a dimension of 2×2. Image courtesy of www.mathsisfun.com.

A great video detailing this is below:

(PLEASE VIEW “MATH IMAGE RESOURCES” PAGE FOR VIDEO SOURCE)

How do we go about multiplying two matrices? Well, this is where things get a little more difficult, but not painfully so! It is often called taking the “dot product” of two matrices, and the beauty is that the two matrices don’t have to be the same dimensions! To save some explanation, let’s actually see it visually:

See above? What happens is we multiply 1 by 7, 2 by 9, and 3 by 11, then sum up the answers to get 58. Notice the first matrix has dimension of 3×2 and the second one 2×3. Image courtesy of https://www.mathsisfun.com/algebra/matrix-multiplying.html

 

Some rules of thumb for matrix multiplication:

  • The m of the first matrix has to equal the n of the of the second. So for above, the m of the first matrix is 2, and the n of the second is 2.
  • The resulting matrix (on the right, above) will have these two numbers as it’s dimensions. So the m from the first, and the n from the second: a 2×2 matrix results.

A great video of this is below:

(PLEASE VIEW “MATH IMAGE RESOURCES” PAGE FOR VIDEO SOURCE)


 

WRAPPING UP AND MOVING ON!

So, what does all of this have to do with tensors?

In my words, a tensor is a mathematical object, similar to a vector or matrix, and often consisting of them. But let’s save all of this for the next section, shall we?

CLICK BELOW TO MOVE ON TO SOME MORE COMPLICATED VECTORS!

…More on Vectors…