If certain entries of this matrix are all nonzero, show that the only

[s3a]

Homework Statement

Matrix U is the following matrix:
http://www.wolframalpha.com/input/?i={{a,b,c},{0,d,e},{0,0,f}}

Question:
If a, d, f are all nonzero, show that the only solution to Ux = 0 is x = 0. Then the upper triangular matrix U has independent columns.

The solution says:
x_complete = x_particular + x_homogeneous = (1/2,0,1/2,0) + x₂ (-3,1,0,0) + x₄(0,0,-2,1)

Homework Equations

x_complete = x_particular + x_homogeneous

The Attempt at a Solution

I'm very confused; I'm not sure where to start.

Any help in figuring out what I need to do would be really appreciated!

 

[Dick]

Homework Statement

Matrix U is the following matrix:
http://www.wolframalpha.com/input/?i={{a,b,c},{0,d,e},{0,0,f}}

Question:
If a, d, f are all nonzero, show that the only solution to Ux = 0 is x = 0. Then the upper triangular matrix U has independent columns.

The solution says:
x_complete = x_particular + x_homogeneous = (1/2,0,1/2,0) + x₂ (-3,1,0,0) + x₄(0,0,-2,1)

Homework Equations

x_complete = x_particular + x_homogeneous

The Attempt at a Solution

I'm very confused; I'm not sure where to start.

Any help in figuring out what I need to do would be really appreciated!

Write out the equations that correspond to Ux=0, putting x=(x1,x2,x3). I think the solution you quoted is for a different problem. U is 3x3 and the solution has four dimensional vectors in it.

 

[s3a]

Oh, oops.

Ux = 0

{{a,b,c},{0,d,e},{0,0,f}} {{x_1},{x_2},{x_3}} = {{0},{0},{0}}

I can see how this gives z = 0 (since f is the nonzero coefficient of in fz = 0) but, I'm confused for the rest.

 

[Dick]

Oh, oops.

Ux = 0

{{a,b,c},{0,d,e},{0,0,f}} {{x_1},{x_2},{x_3}} = {{0},{0},{0}}

I can see how this gives z = 0 (since f is the nonzero coefficient of in fz = 0) but, I'm confused for the rest.

Put z=0 into the other equations. What do they become?

 

[s3a]

Oops. I now see that x = y = z = 0. :)

So, because x = y = z or x_1 = x_2 = x_3 (depending on the notation chosen) shows that the vector x = 0 and the problem is solved, right?

 

[Dick]

Oops. I now see that x = y = z = 0. :)

So, because x = y = z or x_1 = x_2 = x_3 (depending on the notation chosen) shows that the vector x = 0 and the problem is solved, right?

Yes.

 

[s3a]

Yay. :)

Thanks.