Register to reply

Linear algebra question..

by transgalactic
Tags: algebra, linear
Share this thread:
transgalactic
#1
Dec9-08, 03:17 AM
P: 1,398
i got a set of linear equations with x,y,z,t variables.

x=4,y=-2,z=-2,t=4 is a solution
x=-2 y=4 z=4 t=-2 is a solution


x=2,y=2,z=2,t=2 is not a solution.

prove that this set is not homogeneous ?
prove that x=0,y=2,z=2,t=0 is a solution?

??
Phys.Org News Partner Science news on Phys.org
Sapphire talk enlivens guesswork over iPhone 6
Geneticists offer clues to better rice, tomato crops
UConn makes 3-D copies of antique instrument parts
Defennder
#2
Dec9-08, 04:07 AM
HW Helper
P: 2,616
I'm not clear as to what homogenous here means. On the other hand, if a set of linear equations has more than one solutions, then it has infinite number of solutions. That allows you to come up with a general solution given just two specific possible solutions. This should help you with part 2.
HallsofIvy
#3
Dec9-08, 07:21 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,338
Quote Quote by transgalactic View Post
i got a set of linear equations with x,y,z,t variables.

x=4,y=-2,z=-2,t=4 is a solution
x=-2 y=4 z=4 t=-2 is a solution


x=2,y=2,z=2,t=2 is not a solution.

prove that this set is not homogeneous ?

prove that x=0,y=2,z=2,t=0 is a solution?

??
I believe that by "homogeneous" here, you mean that the equations are of the form
[itex]a_1x+ b_1y+ c_1z+ d_1t= 0[/itex]
[itex]a_2x+ b_2y+ c_2z+ d_2t= 0[/itex]
[itex]a_3x+ b_3y+ c_3z+ d_3t= 0[/itex]
[itex]a_4x+ b_4y+ c_4z+ d_4t= 0[/itex]
That is the same as the matrix equation
[tex]\left[\begin{array}{cccc}a_1 & b_1 & c_1 & d_1 \\a_2 & b_2 & c_2 & d_2 \\a_3 & b_3 & c_3 & d_3 \\a_4 & b_4 & c_4 & d_4 \end{array}\right]\left[\begin{array}{c} x \\ y \\ z \\ t\end{array}\right]= \left[\begin{array}{c}0 \\ 0 \\ 0 \\ 0\end{array}\right][/tex]

Such an equation will have a unique solution if and only if the coefficient matrix is invertible. Since this has at least two solutions, the coefficient matrix is not invertible. Since there are at least two solutions, the coefficient matrix is not invertible. If it were homogeneous, then any linear combination of solutions would also be a solution: if [itex]Ax_1= 0[/itex] and [itex]Ax_2= 0[/itex] then [itex]A(cx_1+ dx_2)= cAx_1+ dAx_2= 0[/itex]

transgalactic
#4
Dec9-08, 08:24 AM
P: 1,398
Linear algebra question..

how to prove that
x=0,y=2,z=2,t=0 is not a solution?
Defennder
#5
Dec9-08, 08:49 AM
HW Helper
P: 2,616
You're supposed to prove it's a solution, you mean.
transgalactic
#6
Dec9-08, 12:36 PM
P: 1,398
i am supposed
but i dont know how??
Mark44
#7
Dec9-08, 01:45 PM
Mentor
P: 21,216
You said you have a set of linear equations (which you didn't show). Replace x, y, z, and t with 0, 2, 2, and 0, respectively, on the expressions on the left side of the equations you have. You should get 0 on the right side of all of your equations.
transgalactic
#8
Dec9-08, 02:38 PM
P: 1,398
i got a crazy idea
if i use this given sentences and transform them into equations:
x=4,y=-2,z=-2,t=4 is a solution
x=-2 y=4 z=4 t=-2 is a solution

(4,-2,-2,4)+2*(-2,4,4,-2)=(0,6,6,0)=3*(0,2,2,0)

so i got my solution that i suppose to prove from algebraic manipulation

is that a proove??
if it is what formal words do i need to say in order to confirm this method?
Mark44
#9
Dec9-08, 02:52 PM
Mentor
P: 21,216
Sort of.
Here's the situation as I see it (necessariy sketchy, since you didn't provide many details):
You have a system of equations which I will represent as a matrix equation:
Ax = 0

Any vector x in R^4 is a solution to the equation above if A times x equals the zero vector.

You are given that for x_1 = (4, -2, -2, 4)^T and x_2 = (-2, 4, 4, -2),
A*x_1 = 0 and A*x_2 = 0.

For x_3 = (0, 2, 2, 0), you have shown that x_3 is a linear combination of x_1 and x_2, namely x_3 = 1/3*x_1 + 2/3*x_2.
By the linearity of matrix multiplication A*x_3 = A*(1/3*x_1 + 2/3*x_2)
You should be able to fill in what's missing at the end to show that x_3 is a solution to the homogeneous equation Ax = 0.


Register to reply

Related Discussions
Linear algebra - inner product and linear transformation question Calculus & Beyond Homework 0
Linear Algebra question Precalculus Mathematics Homework 7
Linear Algebra question Calculus & Beyond Homework 1
Another Linear Algebra Question Introductory Physics Homework 1
Linear algebra question Introductory Physics Homework 7