Suppose that you are given the following system of linear equations

[variables are x1, x2, x3 and x4]

x1 + 3x2 + 9x3 + 2x4 =5
x1 + 3x3 - 4x4 =5
x2 + 2x3 + 3x4 =-1
-2x1 + 3x2 + 5x4 =-1

Let b be the vector such that B = [A b] , and let a1, a2, a3 and a4 be the columns of A.

a) What is the general solution for the system?
i just want to double check my answers, please point out any mistakes.

the solution i got is
x1=1-3x3
x2=2-2x3
x3 is free
x4=-1

Let m be the number of linearly independent columns of A, let k be the number of parameters (free variables) in (a), and let n be the total number of columns in A. In our example above, n = 4.

b) Notice that m + k = n. Do you suppose that this relationship will be true for all systems of linear equations? Why or why not?

-i don't know how to approach this question, anybody mind explaining the answer to me? thanks.

x3 is zero, so if one puts that into what you got for x1, x2, you are right. I don't agree with your x3 is "free". It is zero. x3 is not a free parameter.

http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi

I don't understand the b) either.

but the row reduced matrix is the following:

1 0 3 0 1
0 1 2 0 2
0 0 0 1 -1
0 0 0 0 0

with pivots in columns 1, 2 and 4, does that make x3 a free variable?

Yes, I found the same solution. x3 is free so you have a "one dimensional solution space". The rank of the matrix is the number of independent rows (or columns, row rank and column rank are always the same) and this is 3.

what exactly is independent rows?

so as you said, m=3 and k=1
m+k=4, so this is true.

will this be true for all systems of linear equations (as the question asks)? How would you actually explain or proof it?

For b) see here:

http://en.wikipedia.org/wiki/Rank-nullity_theorem

If you write the equation as:

A x = b

if x and y are two solutions:

A x = b

A y = b

then

A (x-y) = 0

So, the difference is an element of the null space and all solutions are of the form:

x = n + p

Where p is one particular solution of:

A x = b

and n is an element of the null space.

The number of free parameters in te solution will thus be equal to the dimension of the null space. The equation m + k = n is thus a direct consequence of the Rank-nullity theorem.

Sketch of the proof here:

http://www.jiskha.com/display.cgi?id=1179363311.1179366947

Thanks, but i haven't learn Rank-nullity theorem or null space yet. We just got introduced to linearly independent and linearly dependent. Is there a way of explaining it with the limited knowledge i have?

Ok,

let's just start with the equation:

A x = b (1)

where A is an N by N matrix, and b is an N dimensional vector. Suppose that a solution exists, let's denote it by x.

Now let's consider the equation:

A y = 0 (2)

If a1 a2, a3, are the columns of A (considered as vectors) and y1, y2, y3, etc. are the components of the vector y, then this equation is simply:

a1 y1 + a2 y2 + a3 y3 + ...=0

Clearly if all the columns are independent then (by definition) you only have the trivial solution y = 0.

If the number of independent columns is k, then that means that you'll have a solution for y that depends on N - k independent parameters. I guess you just have to check in your lecture notes how the concept of "number of independent columns" or "rank" was explained to you.

You can now construct the general solution of (1) by adding to x a solution to (2):

A (x + y) = A x + A y = b + 0 = b

The number of free parameters in y is N - k. So all we have to show now is that there aren't any more degrees of freedom apart from the choice of y, or put differently, that all solutions are of the form x + y for one particular fixed x. But that's trivial. Suppose that z is any solution of (1) and that z is not equal to x. Then the difference of z and x is mapped by A to zero, so z - x would satisfy Eq. (2).

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Therefore, all solutions of (1) can be written as x + y, where x is a particular solution and y is any solution of (2). This means that the number of free parameters in the solution is equal to the dimension of the null space of A, which is N - k.

In conclusion, for any system of linear equations, the equation m + k = n holds true, where m is the number of linearly independent columns of A, k is the number of free parameters in the solution, and n is the total number of columns in A. This relationship is a direct consequence of the Rank-nullity theorem.

To summarize, the general solution for the given system of linear equations is:

x1 = 1-3x3
x2 = 2-2x3
x3 is not a free parameter, it is equal to 0
x4 = -1

In this case, m = 3 (number of independent columns), k = 1 (number of free variables), and n = 4 (total number of columns in A). The equation m + k = n is true in this example, but is it true for all systems of linear equations?

To answer this question, we can refer to the Rank-nullity theorem. According to this theorem, the number of independent columns (m) plus the number of free variables (k) in the solution is equal to the total number of columns (n) in matrix A.

In other words, m + k = n is a direct consequence of the Rank-nullity theorem. This relationship holds true for all systems of linear equations, regardless of the specific values or variables involved.

If you haven't learned about the Rank-nullity theorem yet, you can still understand this concept by considering the fact that the number of independent columns (m) represents the number of linearly independent equations in the system. This means that there are m equations with unique solutions.

The number of free variables (k) represents the additional variables that can take on any arbitrary value. These variables do not contribute to the uniqueness of the solution. In other words, they are "free" to vary.

By adding the number of independent columns (m) and the number of free variables (k), we obtain the total number of columns (n) in matrix A. This relationship holds true for any system of linear equations.

I hope this explanation helps! Let me know if you have any further questions.