# Components of a covariant vector

## 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.

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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 $$j = 1,2,3$$.

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?

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

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?

Does anyone else have insight on this problem?