diff --git a/src/3_vector_spaces.md b/src/3_vector_spaces.md index 2e13ec3920ef1c18dc8ac2f0c0d23bc66ef08dda..4424858d914d9605b18c79d541191d959b465b3c 100644 --- a/src/3_vector_spaces.md +++ b/src/3_vector_spaces.md @@ -90,9 +90,9 @@ You might be already familiar with the concept of performing a number of various !!! info "Scalar multiplication" I can multiply a vector by a scalar number (either real or complex) to produce another vector, $$\vec{c} = \lambda \vec{a}.$$ Addition and scalar multiplication of vectors are both *associative* and *distributive*, so the following relations hold - $$1. \hspace{5pt} (\lambda \mu) \vec{a} = \lambda (\mu \vec{a}) = \mu (\lambda \vec{a})$$ - $$2. \hspace{5pt} \lambda (\vec{a} + \vec{b}) = \lambda \vec{a} + \lambda \vec{b}$$ - $$3. \hspace{5pt} (\lambda + \mu)\vec{a} = \lambda \vec{a} +\mu \vec{a}$$ + $$\begin{align} &1. \qquad (\lambda \mu) \vec{a} = \lambda (\mu \vec{a}) = \mu (\lambda \vec{a})\\ + &2. \qquad \lambda (\vec{a} + \vec{b}) = \lambda \vec{a} + \lambda \vec{b}\\ + &3. \qquad (\lambda + \mu)\vec{a} = \lambda \vec{a} +\mu \vec{a} \end{align}$$ ### Vector products