Components of a covariant vector

[roldy]

Homework Statement

Consider the following two basis sets (or triads) in $${R}^3$$:

$$\{\vec{e}_1, \vec{e}_2, \vec{e}_3\} := \{(1, 0, 0), (0,1, 0), (0, 0, 1)\}$$

$$\{\widehat{\vec{e}_1}, \widehat{\vec{e}_2}, \widehat{\vec{e}_3}\} := \{(1, 0, 0), (1,1, 0), (1, 1, 1)\}.$$

Let a covariant vector $$\tilde{u}$$ be defined by $$\tilde{u}(\alpha^i{\vec{e}_i} ):= \alpha^3-\alpha^2$$. Obtain explicitly the components of $$\tilde{u}$$ relative to the corresponding bases $$\{\vec{e}_1, \vec{e}_2, \vec{e}_3\}$$ and $$\{\widehat{\vec{e}_1}, \widehat{\vec{e}_2}, \widehat{\vec{e}_3}\}$$.

Homework Equations

$$\tilde{a}(\vec{b})=\tilde{a}(\beta^j\vec{e}_j):=\alpha_j\beta^j$$

The Attempt at a Solution

My attempt at a solution is just me running around in a bunch of directions. I really don't have a clear understanding of how to approach this.

 

[ystael]

Start by computing $$\tilde{u}(\vec{e}_j)$$ for $$j = 1,2,3$$. (These are the components of $$\tilde{u}$$ relative to the basis $$\{\vec{e}_1, \vec{e}_2, \vec{e}_3\}$$, or more properly speaking, relative to the corresponding dual basis.)

Then expand the new basis vectors $$\widehat{\vec{e}_j}$$ in terms of the $$\vec{e}_k$$, and use that to compute $$\tilde{u}(\widehat{\vec{e}_j})$$ for $$j = 1,2,3$$.

 

[roldy]

I'm actually confused at how to do the first part, calculating $$\tilde{u}(\vec{e}_j)$$.
I know I need to use this relationship $$\tilde{u}(\alpha^i{\vec{e}_i} ):= \alpha^3-\alpha^2$$.

So expanding the relationship:

$$\tilde{u}(\alpha^1\vec{e}_1+\alpha^2\vec{e}_2+\alpha^3\vec{e}_3)=\alpha^3-\alpha^2$$

What does the above actually mean?

I asked my professor about this problem and he helped a little. He actually worked through a portion of it. This is what came about.

$$\tilde{u}(\vec{e}_1)=\tilde(u)\left((1)\vec{e}_1+(0)\vec{e}_2+(0)\vec{e}_3\right)=0-0=0$$

The thing I don't understand about this is why are there basis $$\vec{e}_1, \vec{e}_2, \vec{e}_3$$ in the parenthesis when you are trying to figure out $$\tilde{u}(\vec{e}_1)$$.

I need a really simplified (dumbed down) explanation on this. I'm not used of working with anything so abstract.

Would you know of any books that are easy on the reader?

 

[roldy]

Does anyone else have insight on this problem?