I'm not sure what your notation means, could you explain it? Is this what you have:
\(\displaystyle \langle \hat e_j | \textbf{a}\rangle = a_j\)?
edit:Okay, I see, in your book: \(\displaystyle a=a_1\hat e_1 + ... + a_n\hat e_n\). The inner product is bilinear:
\(\displaystyle \langle \hat e_j | \textbf{a}\rangle = \langle \hat e_j | a_1\hat e_1 + ... + a_n\hat e_n\rangle\)
\(\displaystyle = a_1\langle \hat e_j | \hat e_1\rangle + .. + a_n\langle \hat e_j | \hat e_n\rangle\).
But \(\displaystyle \langle \hat e_j | \hat e_k\rangle = \delta_{jk}\) (1 if j=k, 0 otherwise).. So:
\(\displaystyle a_1\langle \hat e_j | \hat e_1\rangle + .. + a_n\langle \hat e_j | \hat e_n\rangle = (a_1)0 + ... + (a_j)(1) + ... + (a_n)0 = a_j\)
For example, take the standerd basis of \(\displaystyle \mathbb{R}^4\), and the inner product the standard dot product. Then \(\displaystyle \hat e_3 = (0,0,1,0)\), and let \(\displaystyle a=(1,2,3,4)\). Then we can write \(\displaystyle a = 1\hat e_1 + 2\hat e_2 + 3\hat e_3 + 4\hat e_4\), and:
\(\displaystyle \hat e_3 \cdot a =1(0)+2(0)+3(1)+4(0) = 3 = a_3\).
