1) Why express leading variable in terms of free variables?

2) I know free variables and parameters are not the same. But whenever my textbook shows an example, the free variables and parameters are identical in terms of alphabet representation. Is there ever a time where they are not identical in terms of alphabet representation?

3) What is the whole point of a solution set and writing the free variables as leading variables and writing free variables as itself and then listing the parameters towards end of the solution set?

4) I'm up to G = P + H (General Solution = Particular Solution + Homogeneous Solution)

I have not grasp the reason for labeling a vector notation as a particular solution. The particular solution seems to be a vector notation that is isolated from the free variables. Is that all it is for?

All of these questions don't make a lot of sense without context. I am guessing you are learning about how to solve systems of linear equations, and you don't have examples in mind of why you may want to do that. Linear algebra is a very important subject that you usually only truly understand a couple of years after studying it, when you see that it keeps showing up in many other parts of math and something finally "clicks" in your head. It's probably the case for a lot of people that they forget about the subject before this happens, and that is a shame. Linear algebra is an integral part of how I think about the world, and I use it everywhere.

If I were teaching the course, I would try to start with some examples that would motivate the material that follows. But that's not how the subject is usually taught.

Given all that, I'll try to answer:

1) That's a way to find a parametric solution, which means that you'll be able to generate all the solutions by giving the free variables arbitrary values and computing the other variables.

2) Describing the solution space using free variables is only one possible parametric description. If the solution space has dimension k, any linearly independent set of k vectors can be used to describe the solutions, as a linear combination of the k vectors. The coefficients that go into the linear combination would then be the parameters. The notion of "parameter" is also useful in situations beyond systems of linear equations, where "free variable" is a pretty meaningless notion. For instance, the solutions to x^2+y^2=1 can be described as (cos(alpha), sin(alpha)), with alpha being the parameter.

3) Isn't this the same as question 1?

4) And the question is...

I don't understand your last line. What do you mean by "a vector notation"?

Yeah I'm a bit confused too at first I thought it was a question about logic (since that's where I usually hear term free variable) but it does seem it is about either solving non homogeneous linear equations (i.e Ax = d for matrix A and vector d, with x the variable vector) or differential equations using particular integral/complementary function ;)

Linear algebra is a very important subject that you usually only truly understand a couple of years after studying it, when you see that it keeps showing up in many other parts of math and something finally "clicks" in your head.

Dang! Really. That is a real shame. I was hoping to have a deep understanding of the subject without taking other classes that would use it...Tuition is pretty high and well I like studying by myself. I hate college lectures...bane of my life *sigh*

The textbook says free variables are variables that are not leading variables. This is when the author goes over system of linear equations and Guass elimination.

As for question #4, the author gave an example where he solved the problem that has system of linear equation and got a solution set out of this system. He wrote the solution set in vector notation form. Vector notation is denoted as Really Big para- theses around the variables of the system of linear equations. As for the purpose of a vector notation, I have no clue why it is used.

I am surprised you never heard of free variables or vector notation. I guess I have a really weird Linear Algebra textbook...

As for question #4, I guess the better question to ask is why does this hold true: General Solution = Particular Solution + Homogeneous Solution.

Linear Algebra sure has a lot of terminology and concepts to understand before I can even do the homework.

I apologize for any confusion in my post. Hope this post is more clear. Thanks for answering the first three questions.

Any tips for succeeding in Linear Algebra even if I study this subject by myself? This textbook loves to use lemma and proof and I do not think it is the best way to go to teach a reader to accept the fact. I do like the examples to understand how to get the answers. But from my experience, I know getting the answers is not the whole point of math.

Why can you denote that equation x^2+y^2 = 1 with trig? Does it only apply to that equation you have there?

I am not sure I can answer why the general solution can be found as a particular solution plus the homogeneous solution without introducing a bunch of names of things that might confuse you further, but I'll try.

In order to avoid messy notation, let's think of an example with three variables:

3*x - 5*y + z = 3

2*x + y = 5

Now imagine you have two solutions to that system of linear equations, (x1, y1, z1) and (x2, y2, z2). These x1, y1, z1, x2, y2 and z2 are specific values, not variables. You can actually think of them as x1=2, y1=1, z1=2, x2=1, y2=3, z2=15 [EDIT: I had z2=12, which is incorrect]. If you plug x=x1, y=y1 and z=z1, you get that the first line evaluates to 3 and the second one to 5, as expected. Same thing for x=x2, y=y2 and z=z2. Now, what happens if you plug in x=x2-x1, y=y2-x1, z=z2-z1? Well, the first line is now

3*(x2-x1) - 5*(y2-y1) + (z2-z1) = (3*x2 - 5*y2 + z2) - (3*x1 - 5*y1 + z1) = 3 - 3 = 0

The second one is

2*(x2-x1) + (y2-y1) = (2*x2 + y2) - (2*x1 + y1) = 5 - 5 = 0

So the difference between two solutions is a solution to the homogeneous problem. Similarly, if you have a particular solution and you add a solution to the homogeneous problem, you'll get another solution to the original problem.

Does that make it any more clear?

I am surprised you never heard of free variables or vector notation. I guess I have a really weird Linear Algebra textbook...

Of course I have heard of both. Well, "free variable" is not a very fundamental notion in mathematics: It's more like a peculiarity of one method to solve systems of linear equations, but it's not a term you will encounter very often outside Gaussian elimination.

Vector notation is used everywhere and you should make yourself as familiar with it as possible. The reason for not understanding the last line of your original post is that you are using "a vector notation" to mean something other than what I understand by that. To me "vector notation" means "the convention of describing the settings of several variables at the same time by listing them in some order with parentheses around them", or something of that sort. With that definition in mind, try to re-read the last line in the original post, and you tell me if it makes any sense.

Any tips for succeeding in Linear Algebra even if I study this subject by myself? This textbook loves to use lemma and proof and I do not think it is the best way to go to teach a reader to accept the fact. I do like the examples to understand how to get the answers. But from my experience, I know getting the answers is not the whole point of math.

Try to find examples where you can make use of what you are learning. Since you are on this forum, you are probably interested in game development, and there are many opportunities to use linear algebra in game development, especially in 3D graphics (but also in physics, algorithm analysis, image processing, audio processing...).

Why can you denote that equation x^2+y^2 = 1 with trig? Does it only apply to that equation you have there?

I am not sure what you mean by "denote" here, but the Pythagorean trigonometric theorem (sometimes referred to as the fundamental theorem of trigonometry) says that cos(x)^2 + sin(x)^2 = 1, and the unit circle is a common tool to think about trigonometry. So using cosine and sine as a parametrization of the circle is very natural.

3*(x2-x1) - 5*(y2-y1) + (z2-z1) = (3*x2 - 5*y2 + z2) - (3*x1 - 5*y1 + z1) = 3 - 3 = 0

hmm...wait a second shouldn't the last part be 0 - 3 instead of 3 - 3 ?

Because plugging it: 3(1) - 5(3) + 12 = 3 - 15 + 12 = 0

True. Re-reading it now the statement does feel ambiguous.

I should rewrote question#4 here,

What I mean to say have you ever seen a particular solution written as a vector notation representing values of the leading variables? The reason I ask is because I do not know the sole purpose of a particular solution.

hmm...wait a second shouldn't the last part be 0 - 3 instead of 3 - 3 ?

Because plugging it: 3(1) - 5(3) + 12 = 3 - 15 + 12 = 0

Ooops! I meant z2=15. Sorry about that.

Try to find examples where you can make use of what you are learning. Since you are on this forum, you are probably interested in game development, and there are many opportunities to use linear algebra in game development, especially in 3D graphics (but also in physics, algorithm analysis, image processing, audio processing...).

I only done 2D game programming for almost a year now. I self taught myself early January. None of those projects used linear algebra. It only used one kinematic equation from physics.

I have not found how I can apply this subject to game development. Most of the thought process of creating the logic and mechanics from a 2D game stems from programming and basic understanding of 2D graphics.

Any recommendations of linear algebra applications for 2D games?

I will do my own research as well.

P.S. What is the point of a particular solution?

I am not sure what you mean by "denote" here, but the Pythagorean trigonometric theorem (sometimes referred to as the fundamental theorem of trigonometry) says that cos(x)^2 + sin(x)^2 = 1, and the unit circle is a common tool to think about trigonometry. So using cosine and sine as a parametrization of the circle is very natural.

does this theorem only apply for cosx^2 + sinx^2 = 1? I do remember the equation from pre-calculus.

replace denote with "represent".

Similarly, if you have a particular solution and you add a solution to the homogeneous problem, you'll get another solution to the original problem.

When you said solution, do you mean one of the many variables that belongs to a particular solution?