LINEAR ALGEB

Vector and spaces

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms (listed below in § Definition). To specify that the scalars are real or complex numbers, the terms real vector space and complex vector space are often used.

Matrix Transformations

Objectives

1. Learn to view a matrix geometrically as a function.
2. Learn examples of matrix transformations: reflection, dilation, rotation, shear, projection.
3. Understand the vocabulary surrounding transformations: domain, codomain, range.
4. Understand the domain, codomain, and range of a matrix transformation.
5. Pictures: common matrix transformations.
6. Vocabulary words: transformation / function, domain, codomain, range, identity transformation, matrix transformation.

In this section we learn to understand matrices geometrically as functions, or transformations. We briefly discuss transformations in general, then specialize to matrix transformations, which are transformations that come from matrices.

Matrices as Functions

Informally, a function is a rule that accepts inputs and produces outputs. For instance, f(x)=x2 is a function that accepts one number x as its input, and outputs the square of that number: f(2)=4. In this subsection, we interpret matrices as functions.

Let A be a matrix with m rows and n columns. Consider the matrix equation b=Ax (we write it this way instead of Ax=b to remind the reader of the notation y=f(x)). If we vary x, then b will also vary; in this way, we think of A as a function with independent variable x and dependent variable b.

• The independent variable (the input) is x, which is a vector in Rn.
• The dependent variable (the output) is b, which is a vector in Rm.

Matrix Transformations

Now we specialize the general notions and vocabulary from the previous subsection to the functions defined by matrices that we considered in the first subsection.

Definition

Let A

 be an m

×

n

 matrix. The matrix transformation associated to A

 is the transformation

.

This is the transformation that takes a vector

Bases as Coordinate Systems

Objectives

1. Learn to view a basis as a coordinate system on a subspace.
2. Recipes: compute the B
3. -coordinates of a vector, compute the usual coordinates of a vector from its B
4. -coordinates.
5. Picture: the B
6. -coordinates of a vector using its location on a nonstandard coordinate grid.
7. Vocabulary word: B
8. -coordinates