# Proof of linear independence

• Jan 14th 2010, 08:00 PM
Mollier
Proof of linear independence
problem:

(a)
Prove that the four vectors
$\displaystyle x=(1,0,0),\;y=(0,1,0),\;z=(0,0,1)\;u=(1,1,1)$ in $\displaystyle \mathbb{C}^3$ form a linearly dependent set, but any three of them are linearly independent.

(b)
$\displaystyle x(t)=1,\;y(t)=t,\;z(t)=t^2,\;u(t)=1+t+t^2$, prove that $\displaystyle x,y,z,u$ are linearly dependent, but any three of them are linearly independent.

attempt:

(a) To prove that the four vectors are linearly dependent, I guess I could set up four linear equations and solve them.
But I'm thinking it should be enough to show that:
$\displaystyle x+y+z-u=0$.

The thing I do not understand is the "any three of them are linearly independent" part.
How do I go about proving that without checking all combinations?

(b) $\displaystyle x+y+z-u=0$. Again, I don't know how to prove the "any three of them" part without checking all combinations.

Thanks!
• Jan 14th 2010, 09:04 PM
Black
For both (a) and (b), you're pretty much spot on for proving linear dependence.

For (a), x, y, and z being independent is clear. Therefore, we need to show that u and any two of x,y, and z are linearly independent. Let $\displaystyle x=\mathbf{e}_1, y=\mathbf{e}_2, z=\mathbf{e}_3$ and let $\displaystyle i,j \in \{1,2,3\}$, where $\displaystyle i \not= j$. If we have (for $\displaystyle a,b,c \in \mathbb{C}$)

$\displaystyle a\mathbf{e}_i+b\mathbf{e}_j+cu=0$ ,

then no matter what i and j are, we will always end up with the following system of equations:

$\displaystyle a+c=0$

$\displaystyle b+c=0$

$\displaystyle c=0$,

which implies $\displaystyle a=b=c=0$.

Working out part (b) is very similar.

• Jan 14th 2010, 10:37 PM
Mollier
Quote:

Originally Posted by Black

For (a), x, y, and z being independent is clear. Therefore, we need to show that u and any two of x,y, and z are linearly independent.

Can I use the statement that "x,y,z being independent is clear" in a proof?

As for (b), is this ok:

Let $\displaystyle x=t^0,\;y=t^1,\;z=t^2,\;i,j=\{0,1,2\},\;i\neq j$
If for $\displaystyle a,b,c \in \mathbb{C}$:

$\displaystyle at^i+bt^j+cu=0$

then for all i and j we get a=b=c=0.

Thanks.
• Jan 15th 2010, 02:59 AM
Black
Quote:

Originally Posted by Mollier
Can I use the statement that "x,y,z being independent is clear" in a proof?

I guess it depends on the teacher that's grading your work. You should prove that they are independent (to be on the safe side), but it's very straightforward.

Quote:

Originally Posted by Mollier
As for (b), is this ok:

Let $\displaystyle x=t^0,\;y=t^1,\;z=t^2,\;i,j=\{0,1,2\},\;i\neq j$
If for $\displaystyle a,b,c \in \mathbb{C}$:

$\displaystyle at^i+bt^j+cu=0$

then for all i and j we get a=b=c=0.

Thanks.

Yep, pretty much. No matter what i and j are, when you group the like terms together and compare coefficients, you'll end up with the same system of equations as part (a).
• Jan 15th 2010, 03:40 AM
Raoh
Quote:

Originally Posted by Mollier
problem:

(a)
Prove that the four vectors
$\displaystyle x=(1,0,0),\;y=(0,1,0),\;z=(0,0,1)\;u=(1,1,1)$ in $\displaystyle \mathbb{C}^3$ form a linearly dependent set, but any three of them are linearly independent.

(b)
$\displaystyle x(t)=1,\;y(t)=t,\;z(t)=t^2,\;u(t)=1+t+t^2$, prove that $\displaystyle x,y,z,u$ are linearly dependent, but any three of them are linearly independent.

attempt:

(a) To prove that the four vectors are linearly dependent, I guess I could set up four linear equations and solve them.
But I'm thinking it should be enough to show that:
$\displaystyle x+y+z-u=0$.

The thing I do not understand is the "any three of them are linearly independent" part.
How do I go about proving that without checking all combinations?

(b) $\displaystyle x+y+z-u=0$. Again, I don't know how to prove the "any three of them" part without checking all combinations.

Thanks!

hi
$\displaystyle u\in Span(z,y,x)$
and $\displaystyle (z,y,x)$ is a simple basis of $\displaystyle \mathbb{C}^3$ which means it's L.I.
• Jan 15th 2010, 03:47 AM
Raoh
Quote:

Originally Posted by Mollier
Can I use the statement that "x,y,z being independent is clear" in a proof?

As for (b), is this ok:

Let $\displaystyle x=t^0,\;y=t^1,\;z=t^2,\;i,j=\{0,1,2\},\;i\neq j$
If for $\displaystyle a,b,c \in \mathbb{C}$:

$\displaystyle at^i+bt^j+cu=0$

then for all i and j we get a=b=c=0.

Thanks.

hi
put,
$\displaystyle \lambda _0+\lambda _1t+\lambda _2t^2=0$
For $\displaystyle t=0$ ,you get $\displaystyle \lambda _0$.
Differentiate with respect to $\displaystyle t$,
$\displaystyle \lambda _1+2\lambda _2t=0.$
put $\displaystyle t=0$ and get $\displaystyle \lambda _1=0.$
....
...and you get $\displaystyle \lambda _2=0.$
• Jan 15th 2010, 03:57 AM
Mollier
Great stuff guys, thank you very much!

Edit-
Roah: just saw you last post, sweet! :)