0

enter image description here

Okay so I completed part (i) and I got some help on part (ii) so I am fine with that now. I'm stuck on part (iii) though and don't really understand. Any help would be appreciated

user109331
  • 51
  • 1
  • 6
  • Taking the inner product of the unit vector and the vector then multiplying it by the unit vectors in each direction should intuitively lead to $\vec{v}$! For instance, in standard $\hat{i}$ and $\hat{j}$ components, $v=(v\cdot i)\hat{i} +(v\cdot j)\hat{j}=v_i\hat{i}+v_j\hat{j}.$ – William Chang Mar 10 '14 at 05:43

2 Answers2

1

The notation $\mathbb{H} = \operatorname{lin} \{ e_1, ..., e_n \}$ means that any point $v \in \mathbb{H}$ can be written as a linear combination of the $e_k$.

So let $v \in \mathbb{H}$ and write $v = \sum_k x_k e_k$. Then note that $\langle e_i , v \rangle = \sum_k x_k \langle e_i , e_k \rangle = x_i$, since the $e_k$ are orthonormal (and so $\langle e_i , e_k \rangle = \delta_{ik}$).

Hence $v = \sum_k \langle e_i , v \rangle e_k$.

copper.hat
  • 172,524
0

Intuitive explanation: $\forall k, \langle \vec{v},\hat{e_k} \rangle$ is simply the component of $v$ going in the $k$ direction - let this be $v_k$. Thus $\Sigma_{k=1}^n \langle \vec{v},\hat{e_k} \rangle \hat{e_k}=v_1\hat{e_1}+v_2\hat{e_2}+...+v_n\hat{e_n}=(v_1,v_2,...,v_n)=\vec{v}.$

In Cartesian Tensor Notation, we simply write this as $\vec{A}=A_j$, with an implied sum running from $j=1$ to $j=n$.

I hope this helps you understand the problem better and see why it's true!

William Chang
  • 2,369