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MATH 235
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Richard Ennis
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University of Waterloo

Mathematics

MATH 235

Richard Ennis

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.
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