Linear Algebra (Pt. 1): 2x2
Geometry of Linear Equation
We will be looking to solve linear algebra for n linear equations with n unknowns. There are 2 ways to express linear equations in terms of matrices, namely
Row Picture
Column Picture
Row Shorthand
We will take reference to the above 2x2 problem, before forward to 3x3 subsequently.
It may also be helpful to lay out the common terminologies used in the video lecture. Dr. Strang would refer to
A as the coefficient matrix
x as the variable matrix, and
b as the constant matrix
As would the name imply, the row picture merely expresses each row of the linear equation into matrices. Thus, there are no explicit separation between each variables. Each line of equation still stands as it were,
(2x - y = 0) for the first equation
(-x + 2y = 3) for the second equation
Row Picture
Hence, if we were to express this in a picture form with these equations, we would merely obtain lines.
Do note an important fundamental concept of:
Multiple x and y values are able to satisfy each individual equation (hence we are able to draw a line)
But only one x and y value satisfies both equations simultaneously (the point of intersection 1, 2)
Hence, the solution to the above problem is x = 1, y = 2.
Column Shorthand
Dr. Strang loves his column picture.
In the column picture, there will no longer be an explicit "variable matrix", as each coefficient matrix are instead designed to fit for each individual variable.
In other words, the linear equations are separated out into its individual columns (variables) in the form of matrices for the column picture.
As opposed to the row picture, there aren't multiple solutions of x and y to draw a line for each equation. There only exists co-ordinates for each variable, similar to vectors.
Column Picture
Hence, we can attempt to draw these 3 vectors out to observe its column picture.
The big question to this would be "How many combinations of x and y would we require to reach the point of result?"
As you would have known from the Row Picture, the solution is x = 1, y = 2. However this time, the solution is obtained through the right combination of vectors. In Mathematics, this is also termed as finding the right Linear Combination.
Side-note
Another interesting way to perform matrix multiplication was used as well in the lecture. Personally, I feel this is slightly less intuitive for me, and hence would not put this under the Matrix Multiplication" page, but I believe it is good to know as well!
More on the next page ⏭
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