Linear Algebra (Pt. 2): 3x3

3x3 Linear Equations

It feels I have written a lot, but still have yet to cover 3x3 problems (fml) 👵.

oof

This is also to highlight the non-linear increase in complexity to solve these equations with every increase in dimension.

In a 3x3 problem, we will require 3 equations with 3 variables (unknowns) to proceed. The following problem is extracted from video lecture 1 of MIT 18.06 on YouTube.

$$2x - y = 0\\ -x=2y-z=-1\\ -3y+4z=4$$

Row Shorthand

Sticking to the abbreviations used by Strang, the linear equations can be viewed as:

As mentioned previously, the equations for the row picture is left intact. Hence, the coefficient matrix A is structured in a way such that each row are the coefficients of each linear equation.

Row Picture

As now there are 3 unknowns (variables), we would require an additional axis and the picture should be a 3-D one.

Row Picture of 3x3 problem

You will have to excuse my drawing here (despite having a computer lol), as I am not Rembrandt as well 🤪. We can observe that

• Each equation has solutions that are essentially a 2-D plane

• While considering 2 planes, its simultaneous solution is essentially a line

• While considering 3 planes, its simultaneous solution is essentially a point,

all while bearing that each solution to an equation should not be parallel.

As you see, we cannot easily solve for the point, the solution that simultaneously satisfy all 3 equations. Thus, the row picture proves to be reliable only up till 2x2 problems.

Column Shorthand

As mentioned in the column picture, the structure of each equations are broken up into its columns (variables). Therefore, the coefficient matrix will describe each unknown (variable), instead of individual equations. The column shorthand is as of follows:

Column Shorthand for 3x3

We know the solution is probably z = 1 as Dr. Strang has specially designed z's coefficient matrix to equate to the result. This is purely illustrating for the purpose of understanding instead of solving (use of elimination method @ next part 👵).

Column Picture for 3x3

Similar to the column picture of 2x2, the coefficient matrices merely represents vectors in 3-D space. Since Col 3 (z) equals to Col 4 (b or RHS), there is no need to go through the whole trouble of finding the linear combination of vectors like 2x2 (thank god 🥵).

More on the next page ⏭