"Extending by linearity" - meaning?

Question

In Sagan's book The Symmetric Group, he introduces the inner product on $\mathbb{C}\{\mathbf{1}, \mathbf{2}, \mathbf{3}\}=\left\{c_1 \mathbf{1}+c_2 \mathbf{2}+c_3 \mathbf{3} \mid c_{1} c_{2}, c_{3} \in \mathbb{C}\right\}$ by $$ \langle\mathbf{i}, \mathbf{j}\rangle=\delta_{i, j} $$ for any two vectors in the basis $\{\mathbf{1},\mathbf{2},\mathbf{3}\}$. He then writes "Now we extend by linearity in the first variable and conjugate linearity in the second to obtain an inner product on the whole vector space". Can someone explain what "extend by linearity" means in this case?

Answer 1

Since $\{ \mathbf{1}, \mathbf{2}, \mathbf{3} \}$ is a basis for $\mathbb{C}\{ \mathbf{1}, \mathbf{2}, \mathbf{3}\}$, every vector $x \in \mathbb{C}\{ \mathbf{1}, \mathbf{2}, \mathbf{3}\}$ can be represented as $x = a \mathbf{1} + b \mathbf{2} + c \mathbf{3}$ for some coefficients $a, b, c \in \mathbb C$. This expression is linear in the coefficients.

Because you know how a scalar product interacts with the linear structure, you can now deduce the value of the scalar products for any vectors in $\mathbb{C}\{ \mathbf{1}, \mathbf{2}, \mathbf{3}\}$, e.g. $$ \langle a \mathbf{1}, \mathbf{2} \rangle = a \langle \mathbf{1}, \mathbf{2} \rangle, \qquad a \in \mathbb C $$ or $$ \langle \mathbf{1} + b \mathbf{3}, \mathbf{2} \rangle = \langle \mathbf{1}, \mathbf{2} \rangle + \overline{b}\langle \mathbf{3}, \mathbf{2} \rangle, \qquad b \in \mathbb C. $$