This section covers some of the most important concepts in all of linear algebra. With the definitions of vector space and linear maps clearly outlined, the good stuff starts here.
The section introduces the concept of a null space, the concept of a range, and a few other important ones. The most important result from this section is the fundamental theorem of linear maps.
Lastly, system so equations are introduced here. We aim to reframe questions about systems of linear equations into questions about linear maps and then use properties of linear maps to answer those questions.
Important Definitions
The null space of a linear map T, denoted by null T, is the subset of V consisting of those vectors that T maps to 0.
A function ${latex.inlineT: V \rightarrow W} is called injective if Tu = Tv implies u = v. This is also known as a function that is one to one.
For a linear map, the range of T is the subset of W consisting of those vectors that are equal to Tv for some ${latex.inlinev \in V}.
A function ${latex.inlineT: V \rightarrow W} is called surjective if its range equals W. This is also known as a function that is onto.
A homogenous system of linear equations means that the constant term on the right side of each equation is 0.
Relevant Theorems * 1756253268 - Axler 3.13 The null space is a subspace. * 1756253705 - Axler 3.15 A function is injective if and only if the null space is {0} * 1756253793 - Axler 3.18 The range is a subspace. * 1756253933 - Axler 3.21 Fundamental theorem of linear maps. * 1756253983 - Axler 3.22 A linear map to a lower dimensional space is not injective. * 1756254034 - Axler 3.24 A linear map to a higher dimensional space is not surjective. * 1756254061 - Axler 3.26 A homogenous system of linear equations. * 1756254105 - Axler 3.28 An inhomogenous system of linear equations.