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MATH 235 (1)


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University of Waterloo
MATH 235
Richard Ennis

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
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