When an index variable appears twice in a single term and is not otherwise defined, it implies summation of that term over all the values of the index.

$$ y=c_ix^i=\sum_ic_ix^i. $$

column vector

$$ \vec u=e_iu^i=\begin{bmatrix}u^1\\\vdots\\u^k\end{bmatrix} $$

get on column vector, basis is covector and components are contravariant vector

$$ \vec c=\vec u\times \vec v=\varepsilon^i_{jk}u^jv^ke_i=\delta^{il}\varepsilon_{ljk}u^jv^k=c^ie_i $$

$\delta^{ik}$保持分量的逆变性。

观察矩阵、向量乘法，分量保持逆变性。

$$ \vec u\cdot\vec v=u^iv_i=u_jv^j $$

$$ u^i=A^i_jv^j $$

$$ C^i_k=A^i_jB^j_k $$

the outer product：

$$ u^iv_j=(uv)^i_j=A^i_j $$

Raising and lowering indices

Given a tensor, one can raise an index or lower an index by contracting the tensor with the metric tensor, $g_{μν.}$

$$ g_{\mu\sigma}T^\sigma_\beta=T_{\mu\beta} $$