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## Linear algebra

### Unit 1: Lesson 6

Matrices for solving systems by elimination

# Solving linear systems with matrices

Sal solves a linear system with 3 variables by representing it with an augmented matrix and bringing the matrix to reduced row-echelon form. Created by Sal Khan.

## Want to join the conversation?

• This method (converting to reduced row echelon form) seems somewhat incoherent. I'm not understanding the pattern to doing this. How is this possible without unbalancing the equations?
• Basically, you're trying to find one pivotal entry, which must be in their own columns from left to right, while reducing the other coefficients to 0. You can use basic arithmetic to reduce the numbers in the matrix, by performing matrix row operations.
• I wonder if there is a specific method we choose what operations to perform in the matrix when we try to reduce it,like if there is a method for example "first subtract the 1st row from the 2nd,then the 2nd from the multiple of the 3rd by 2 ",etc.I guess probably not,but i had to ask to be sure i m not missing something.
• There is no specific method to simplify any system of three equations. One simply has to look at the equations that one is given and determine how to make the X, Y, and Z zero based on that system's coefficients.
Fascinating question.
• So three separate planes in R3 intersect at a single point in that R3 space. Fair enough.
About the previous video, we had three (presumably) cubes (or three random 3D shapes) in R4, and one way or another, they intersected at a whole plane?

This 4th dimension has boggled my mind for a really long time. I wonder why humans, or at least myself, aren't able to conceptualize it meaningfully.
• Dionys Burger's "Sphereland" provides a good start to 'getting a sense of' the physical 4th dimension - I dare not say visualize! (sequel to another great book "Flatland" by Edwin Abbott)
• Is there a video that introduces the reduced row echelon form ?
• I see Khan is using " = " sign but in my book we aren't allowed to use it between matrices, we use " ~ " instead. Can anyone explain that?
• `a ~ b` usually refers to an equivalence relation between objects `a` and `b` in a set `X`. A binary relation `~` on a set `X` is said to be an equivalence relation if the following holds for all `a, b, c` in `X`:
(Reflexivity) `a ~ a`.
(Symmetry) `a ~ b` implies `b ~ a`.
(Transitivity) `a ~ b` and `b ~ c` implies `a ~ c`.

In the case of augmented matrices `A` and `B`, we may define `A ~ B` if and only if `A` and `B` are augmented matrices corresponding to systems of equations having the same solution set. In this case `~` clearly is an equivalence relation. Since `A` may be different from `B` (they may have the same solution set, but they need not be the same system), writing `A = B` is not strictly correct. In short: use `~`.
• I've progressed through the videos, and don't recall him covering augmentation at any point. What is augmentation exactly? What video does he cover it in?
• An augmented matrix simply means that there's that division between the part you have to reduce and the last column, which is the "answer".
{1+2+3
{3+2+1
would be represented in a "normal" matrix like this:
[1 2 3]
[3 2 1]
while
{1+2+3=6
{3+2+1=6
would have an augmented matrix, with the line instead of the = sign.
[1 2 3 |6]
[3 2 1 |1]
• When he said 3 unknown and 3 equations, did he mean that the x,y,z are the unknowns and the augmented values are the equations?

Please I don't know for sure. I don't want to guess on a quiz.

• You are correct: x, y, and z are the unknowns. The augmented matrix contains the coefficients of the unknowns on the left.
• Is the reduced row echelon form the same as the guass jordan elimination?
• Reduced row echelon form is what Gaussian Elimination achieves. So Gaussian Elimination is the method, reduced row echelon is just the final result.