1. ## Proof!

1. Prove that if u is orthogonal to both v and w, then u is orthogonal to both v + w.
2. Prove that if u is orthogonal to both v and w, then u is orthogonal to sv + tw for all scalar s and t.

i think...
u * (vw) = 0 -> u*v + u*w = 0
u * (v + w) = 0 -> u*v + u*w = 0
.....

2. Hello,
Originally Posted by aeubz
1. Prove that if u is orthogonal to both v and w, then u is orthogonal to both v + w.
2. Prove that if u is orthogonal to both v and w, then u is orthogonal to sv + tw for all scalar s and t.

i think...
u * (vw) = 0 -> u*v + u*w = 0
u * (v + w) = 0 -> u*v + u*w = 0
.....
I don't understand why vw in what you wrote ??

If u is orthogonal to v, it means that u.v=0, where . denotes the dot product.
We have :
u.v=0
u.w=0

What is the dot product of u with v+w ?
u.(v+w)=u.v+u.w, by a property (*) of the dot product.
And this equals 0 by the hypothesis. Hence u is orthogonal to v+w.

Same reasoning for the other one (you'll have to use properties (*) and (**))

------------------------------------------------------------------
(*) $\vec{x} \cdot (\vec{y}+\vec{z})=\vec{x} \cdot \vec{y}+\vec{x} \cdot \vec{z}$

(**) $\forall \lambda \in \mathbb{R} ~,~ \vec{x} \cdot (\lambda \vec{y})=\lambda (\vec{x} \cdot \vec{y})$

3. Another question !!

Thx Moo!

Prove that (u + v) * (u - v ) = ||u||^2 - ||v||^2 for all vectors u and v in R^n.

u.u - u.v + v.u - v.v
u.u - v.v
||u||^2 - ||v||^2

??

And is ||u + v||^2 same thing as (u+v).(u+v)

4. Originally Posted by aeubz
Another question !!

Thx Moo!

Prove that (u + v) * (u - v ) = ||u||^2 - ||v||^2 for all vectors u and v in R^n.

u.u - u.v + v.u - v.v
u.u - v.v
||u||^2 - ||v||^2

??

And is ||u + v||^2 same thing as (u+v).(u+v)
Hmmm basically, we have this :

$||\vec{u}||^2=\vec{u} \cdot \vec{u}$

For any vector u. So it is yes to all your questions ^^

Note that in order to say that -u.v+v.u=0, you have to use the commutative property of the dot product

5. Thx again Moo!

does anyone maybe have a link to prove the properties of the transpose?
Transpose - Wikipedia, the free encyclopedia

My teacher said we should know how to prove these...
In the website link, # 1 , 2 , 3 , 4 , and (A^r)^T = (A^T)^r for all nonnegative integers.

6. Originally Posted by aeubz
Thx again Moo!

does anyone maybe have a link to prove the properties of the transpose?
Transpose - Wikipedia, the free encyclopedia

My teacher said we should know how to prove these...
In the website link, # 1 , 2 , 3 , 4 , and (A^r)^T = (A^T)^r for all nonnegative integers.
You're welcome !

But you'd better post new questions in new threads

7. Originally Posted by Moo
$||\vec{u}||=\vec{u} \cdot \vec{u}$
For any vector u.
It should be: $||\vec{u}||^{\color{red}2}=\vec{u} \cdot \vec{u}$
For any vector u.