- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
1.9 Components
Let $V$ be a linear space of dimension $n$ and consider a basis whose elements $e_1, ..., e_n$ are taken in a given order. We denote such a basis as an $n$-tuple $(e_1, ..., e_n).$ If $x \in V,$ we can express $x$ as a linear combination of these basis elements: \begin{align*} x &= \sum_{i = 1}^n c_ie_i \qquad (1.5) \end{align*} The coefficients in this equation determine another $n$-tuple $(c_1, ..., c_n)$ that is uniquely determined by $x.$ In fact, if we have another representation of $x$ as a linear combination of $e_1, ..., e_n,$ say $\sum_{i = 1}^k d_ie_i,$ then subtracting this from (1.5) we get \begin{align*} \sum_{i = 1}^n (c_i - d_i)e_i &= O. \end{align*} But because the elements $e_1, ..., e_n$ are independent, this implies that $c_i = d_i$ for each $i,$ or in other words: \begin{align*} (c_1, ..., c_n) &= (d_1, ..., d_n) \end{align*} The components of the ordered $n$-tuple $(c_1, ..., c_n)$ determined by equation $(1.5)$ are called the components of $x$ relative to the ordered basis $(e_1, ..., e_n).$