University:
Whitman College
Course:
MATH-240 | Linear Algebra
Academic year:

2022
Views:

252

Pages:

1
Author:

Clark Morrison

Section 4.2 Context: After defining a vector space and subspace in 4.1, we look at premier examples of subspaces associated to an m × n matrix: The Four Fundamental Subspaces of a matrix: The null space, column space and row space of A, and the null space of AT . 1. DEFINITIONS: • Nullspace of A: Null(A) is the set of all x so that Ax = 0. Therefore, it is a subset of the Domain of the mapping x 7→ Ax, IRn . • Rowspace of A: Row(A) is the set of all linear combinations of the rows of A, and is in IRn . • Column space of A: Col(A) is the set of all b so that Ax = b, for x ∈ Domain. Note that the Column space is the Range of the mapping x 7→ Ax, IRm . • Nullspace of AT : Null(AT ) is the set of all b in IRm so that AT b = 0 Therefore, we can think of an m × n matrix as a way of splitting IRn and IRm into 4 pieces (2 pieces each). 2. If the columns of A are linearly independent, Null(A) = {0} 3. If the columns of A span IRm , then Col(A) = IRm . 4. Theorem: If A is m × n matrix, the Nullspace and Column space are subspaces of IRn and IRm , respectively. 5. Relationship between the linear transformation T : IRn 7→ IRm and its matrix representation A: (1) Kernel of T is Null(A) and (2) Range of T is Col(A) 6. Computational considerations: • Is b in Null(A)? Answer by seeing if Ab = 0. • Is b in Col(A)? Answer by solving the augmented system: (A|b) to see if there is an x so that Ax = b. • Describe Col(A): Span {a1 , . . . , an } • Describe Null(A): Solve Ax = 0 and write the answer in parametric (vector) form. 1

Linear Algebra Section 4.2 - Study Guide

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