1. ## Orthogonal projection

I'm unable to do the following problem : Let $V$ be a vector space over $\mathbb{K}$ whose dimension is $n$ and with an inner product $<,>$. Let $W$ be a subspace of $V$ with dimension $m$ and $\{ w_1,...,w_m \}$ be an orthonormal basis of $W$.
We define the orthogonal projection over $W$ as $P_W(v)=\sum_{i=1}^{m} w_i$.
Demonstrate that $||v-P_W(v) || \leq || v-w||$, $\forall w \in W$.
My attempt : I suppose it's true and want to see that it's really true ( don't want to fall over any contradiction).
So $||v-P_W(v) || \leq || v-w||$ $\Leftrightarrow ||v-P_W(v) ||^2 \leq || v-w||^2$ $\Leftrightarrow \leq $. And I didn't reach anything.

2. Originally Posted by arbolis
I'm unable to do the following problem : Let $V$ be a vector space over $\mathbb{K}$ whose dimension is $n$ and with an inner product $<,>$. Let $W$ be a subspace of $V$ with dimension $m$ and $\{ w_1,...,w_m \}$ be an orthonormal basis of $W$.
We define the orthogonal projection over $W$ as $P_W(v)=\sum_{i=1}^{m} w_i$.
Demonstrate that $||v-P_W(v) || \leq || v-w||$, $\forall w \in W$.
My attempt : I suppose it's true and want to see that it's really true ( don't want to fall over any contradiction).
So $||v-P_W(v) || \leq || v-w||$ $\Leftrightarrow ||v-P_W(v) ||^2 \leq || v-w||^2$ $\Leftrightarrow \leq $. And I didn't reach anything.
Okay, here's how I would do it:

Let $\{w_1,...,w_m \} \subset \{v_1,...,v_n \}$ where $w_i=v_i$ for $1 \leq i \leq m$ and $\{v_i \}$ is an orthonormal basis for $V$, and so we have $v= \sum_{i=1}^ n \ {v_i}$ and so if $u=P_W (v)= \sum_{i=1}^ m \ {v_i}$ :

$\Vert v-u \Vert ^2 = \Vert \sum_{i=1}^ n \ {v_i} - \sum_{i=1}^m \ {v_i} \Vert ^2 = \Vert \sum_{i=m+1}^ n \ {v_i} \Vert ^2$

Now, pick $w \in W$ then $w= \sum_{i=1}^ m \ {v_i}$ and so:

$\Vert v-w \Vert ^2 = \Vert \sum_{i=1}^ m \ {v_i} + \sum_{i=m+1}^ n \ {v_i} \Vert ^2$ $= \Vert \sum_{i=1}^ m \ {v_i} \Vert ^2 + \Vert \sum_{i=m+1}^ n \ {v_i} \Vert ^2 \geq \Vert \sum_{i=m+1}^ n \ {v_i} \Vert ^2$ $= \Vert v-u \Vert ^2$

Another thing, in your post you have $\Vert a-b \Vert ^2 = $ but it's actually $ - - + $.

3. Thank you very much Jose. I'll think about it tomorrow. To tell you the truth I was so tired when I posted my thread that I actually didn't think a lot but rather used my short term memory to post what I did today.
I need a rest. My exam is on Monday so I'm pushing hard!
See you!