Linear algebra is the study of vector spaces and special functions between vector spaces called linear maps or linear transformations. Although vector spaces are interesting and useful, the real stars of linear algebra are linear maps.
When you hear something described as “linear”, or hear the phrase “by linearity”, it is just a fancy way of saying that a function distributes over linear combinations of vectors.
A linear map or linear transformation is a function, \begin{equation*} L:V \to W \end{equation*} between two vector spaces \(V\) and \(W\) (the domain and codomain respectively) satisfying for all \(\vec{u}, \vec{v} \in V\) and for all \(c \in \R\text{:}\)
Although the previous definition is often simplest to use, it is common to combine the previous definition's two requirements into a single requirement. That is, we could have defined a map, \(L : V \to W\) to be linear if for all \(\vec{v}_1, \vec{v}_2 \in V\) and for all \(c_1, c_2 \in \R\text{:}\) \begin{equation*} L(c_1 \vec{v}_1 + c_2 \vec{v}_2) = c_1 L\vec{v}_1 + c_2 L\vec{v}_2. \end{equation*} By combining the two requirements into one, it is easier to see that a linear map is just a function which distributes over linear combinations of vectors.
Every linear map sends the zero vector of the domain to the zero vector of the codomain. That is, if \(L : V \to W\) is linear, then \(L\vec{0} = \vec{0}\text{.}\)
The kernel of a linear map, \(L: V \to W\text{,}\) is the subset of all vectors in the domain that get mapped to the zero vector in the codomain. This set is denoted \begin{equation*} \ker(L) = \{\vec{v} \in V : L\vec{v} = \vec{0} \}. \end{equation*}
If \(L:V \to W\) is a linear map, then \(\ker(L)\) is a subspace of \(V\text{.}\)
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