Textbook Notes (362,796)
Mathematics (241)
MATH 235 (1)
Chapter

# Math235.pdf

143 Pages
852 Views
Likes

School
University of Waterloo
Department
Mathematics
Course
MATH 235
Professor
Richard Ennis
Semester
Fall

Description
Chapter 7 Fundamental Subspaces The main purpose of this chapter is to review a few important concepts from the ﬁrst six chapters. These concepts include subspaces, bases, dimension, and linear mappings. As you will soon see the rest of the book relies heavily on these and other concepts from the ﬁrst six chapters. 7.1 Bases of Fundamental Subspaces Recall from Math 136 the four fundamental subspaces of a matrix. DEFINITION Let A be an m × n matrix. The four fundamental subspaces of A are Fundamental m n Subspaces 1. The columnspace of A is Col(A) = {x ∈ R | x ∈ R }. 2. The rowspace of A is Row(A) = {Ax ∈ R | x ∈ R }. 3. The nullspace of A is Null(A) x ∈ R | Ax = 0}. 4. The left nullspace of A is Null(A x ∈ R | A ▯x = 0}. THEOREM 1 Let A be an m×n matrix. Then Col(A) and Null(A ) are subspaces of R and Row(A) n and Null(A) are subspaces of R . Our goal now is to ﬁnd an easy way to determine a basis for each of the four funda- mental subspaces. REMARK To help you understand the following two proofs, you may wish to pick a simple 3×2 matrix A and follow the steps of the proof with your matrix A. 1
More Less

Related notes for MATH 235

OR

Don't have an account?

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Just a few more details

So we can recommend you notes for your school.