Thread: Linear algebra

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

What is the general solution for the system? Write this general solution in
parametric vector form.

i row reduced the augmented matrix into:

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

did i row reduce this matrix correctly?

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

I don't really know the difference between general solution and general solution in parametric vector form, can anybody show me the two? thanks.

Originally Posted by Vidal

the solution i got is
x1=1-3x3
x2=2-2x3
x3 is free
x4=-1
any mistakes?
.

Assuming that you got the row reduced matrix right then your parametrized solutions are correct.

Now let x3 = t where t is any real number to get,

x1 = 1 - 3t
x2 = 2 - 2t
x3 = t
x4 = -1

Originally Posted by Vidal

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

What is the general solution for the system? Write this general solution in
parametric vector form.

i row reduced the augmented matrix into:

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

did i row reduce this matrix correctly?

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

I don't really know the difference between general solution and general solution in parametric vector form, can anybody show me the two? thanks.

TPH gave the general solution, here's the parametric form:

Originally Posted by Vidal

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

What is the general solution for the system? Write this general solution in
parametric vector form.

i row reduced the augmented matrix into:

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

did i row reduce this matrix correctly?

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

I don't really know the difference between general solution and general solution in parametric vector form, can anybody show me the two? thanks.

There is a typo in the reduced augmented matrix in the third row. It should be

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

If the third row were

0 0 0 0 -1

as you have it, it would say 0 = 1 and the equations would be inconsistent with no solutions. Everything else is OK.

thanks everybody. There is another part to this question which i don't know how to do.

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

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