Welcome to our review on
matrices and basic matrix operations.
Before we start talking about
linear systems of ordinary differential equations,
we need to talk about matrices.
A matrix is an m x n array of numbers
where m is a number of rows
and n is a number of columns.
Recall, rows run left to right
and columns run top to bottom.
For example, we denote a three by five matrix as follows.
We often use a capital letter.
Here we have matrix A, which is a three by five matrix.
Again, it's a three by five matrix
because the matrix has three rows,
one, two, three,
and five columns,
one, two, three, four, five.
The numbers as a-ij are called elements or entries.
Where i is the row
and j is the column of the entry or element.
For example, the element a-13 is the element
in row one, column three.
a-24 is the element in row two, column four,
and a-32 is the element in row three column two.
Sometimes you'll also see commas between i and j.
When representing a vector using a matrix,
we usually mean a column vector, meaning an m x 1 matrix.
Remember, this means the matrix has m rows and one column.
If we mean a row vector, we will explicitly say so.
A row vector is a 1 x n matrix,
which means a matrix with one row and n columns.
We usually to know matrices by uppercase letters
and vectors by lowercase letters
with an arrow such as vector x or vector b.
By the zero vector, we mean the vector with all zeros.
Now, to find some properties on matrices.
We want 1 x 1 matrices to really act like numbers
so operations have to be compatible with this viewpoint.
First, we will multiply a matrix by a scaler or number.
We simply multiply each entry in the matrix
by the scaler or number.
For example, here we have two times, a two by three matrix.
To perform the scaler multiplication,
we simply multiply two by each element of the matrix.
Notice two times one is two, two times two is four,
two times three is six, and so on.
Matrix edition is also straightforward.
We add matrices element by element.
For example, here we're adding two two by three matrices.
We simply add the elements in the same position.
One plus one is two.
Two plus one is three.
Three plus negative one is two and so on
and we subtract matrices in the same way.
If the size dimensions do not match up,
then the addition as well as the subtraction
would not be defined.
If we denote by zero, the matrix with all zero entries
where c and d are scalers and A, B, and C are matrices,
we have the following familiar properties
which will remind us of the properties of real numbers.
First, we have a plus zero equals A
which is equal to zero plus A.
This should remind us of the additive identity property.
Next, we have A plus B equals B plus A
which should remind us
of the community of property of addition.
Next, we have the sum of A and B plus C
is equal to a plus the sum of B and C.
This should remind us
of the associative property of addition.
The last two property should remind us
of the distributed property
where c times the sum of A and B
is equal to c times A plus C times B.
Similarly, the sum of c and d times A
is equal to c times A plus d times A.
Another useful operation for matrices is the transpose.
This operation just swaps the rows and columns of a matrix.
The transpose of A is noted as shown here.
As an example, the transpose of this two by three matrix
is equal to this three by two matrix
because we're interchanging the rows in the columns.
Notice the dimensions change from two by three
to three by two.
The first row in the original matrix
becomes the first column in the transpose,
and the second row in the original matrix
becomes the second column in the transpose.
We'll go ahead and stop here for this lesson
and the next lesson we'll talk about matrix multiplication
which is a little bit more involved.
I hope you found this helpful.