In the next linear algebra course you take, the first lecture might be a reminder about what a vector space is (Definition VS), their ten properties, basic theorems and then some examples. The second lecture would likely be all about linear transformations. While it may seem we have waited a long time to present what must be a central topic, in truth we have already been working with linear transformations for some time.

Functions are important objects in the study of calculus, but have been absent from this course until now (well, not really, it just seems that way). In your study of more advanced mathematics it is nearly impossible to escape the use of functions — they are as fundamental as sets are.

Section LT Linear Transformations

Subsection LT: Linear Transformations

Subsection MLT: Matrices and Linear Transformations

Subsection LTLC: Linear Transformations and Linear Combinations

Subsection PI: Pre-Images

Subsection NLTFO: New Linear Transformations From Old

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section ILT Injective Linear Transformations

Subsection EILT: Examples of Injective Linear Transformations

Subsection KLT: Kernel of a Linear Transformation

Subsection ILTLI: Injective Linear Transformations and Linear Independence

Subsection ILTD: Injective Linear Transformations and Dimension

Subsection CILT: Composition of Injective Linear Transformations

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section SLT Surjective Linear Transformations

Subsection ESLT: Examples of Surjective Linear Transformations

Subsection RLT: Range of a Linear Transformation

Subsection SSSLT: Spanning Sets and Surjective Linear Transformations

Subsection SLTD: Surjective Linear Transformations and Dimension

Subsection CSLT: Composition of Surjective Linear Transformations

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section IVLT Invertible Linear Transformations

Subsection IVLT: Invertible Linear Transformations

Subsection IV: Invertibility

Subsection SI: Structure and Isomorphism

Subsection RNLT: Rank and Nullity of a Linear Transformation

Subsection SLELT: Systems of Linear Equations and Linear Transformations

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Subsection LT: Linear Transformations

Subsection MLT: Matrices and Linear Transformations

Subsection LTLC: Linear Transformations and Linear Combinations

Subsection PI: Pre-Images

Subsection NLTFO: New Linear Transformations From Old

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section ILT Injective Linear Transformations

Subsection EILT: Examples of Injective Linear Transformations

Subsection KLT: Kernel of a Linear Transformation

Subsection ILTLI: Injective Linear Transformations and Linear Independence

Subsection ILTD: Injective Linear Transformations and Dimension

Subsection CILT: Composition of Injective Linear Transformations

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section SLT Surjective Linear Transformations

Subsection ESLT: Examples of Surjective Linear Transformations

Subsection RLT: Range of a Linear Transformation

Subsection SSSLT: Spanning Sets and Surjective Linear Transformations

Subsection SLTD: Surjective Linear Transformations and Dimension

Subsection CSLT: Composition of Surjective Linear Transformations

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions

Section IVLT Invertible Linear Transformations

Subsection IVLT: Invertible Linear Transformations

Subsection IV: Invertibility

Subsection SI: Structure and Isomorphism

Subsection RNLT: Rank and Nullity of a Linear Transformation

Subsection SLELT: Systems of Linear Equations and Linear Transformations

Subsection READ: Reading Questions

Subsection EXC: Exercises

Subsection SOL: Solutions