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Lecture 17

Lecture 17.pdf

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Department
Mathematics
Course
MATH 136
Professor
Robert Sproule
Semester
Winter

Description
Wednesday, February 10 − Lecture 17 : Subspaces associated to matrices Concepts: 1. Nullspace. 2. Column-space 3. Row-space 4. Left Nullspace. 17.1 Definition − Let A m × nbe an m by n matrix. The nullspace, Null(A), of the matrix A is defined as follows: Null(A) = {x ∈ ℝ : Ax = 0}. Determining the nullspace of a matrix A essentially comes down to solving a homogeneous system of linear equations. n m If T : ℝ → ℝ is a linear mapping which induces the matrix A then Ker(T) = Null(A). - We know that the solution set S of a homogeneous system of linear equation is of the form S = Span{a , a , …, a }. So the nullspace of the matrix A 1 2 m Null(A) = Span{a , a , …, a } 1 2 m is a subspace. By checking the linear independence of the spanning family {a 1, a2, …, a m we can determine whether this spanning family is a basis of Null(A) or not. 17.2 Definition – Let A m×n = [c 1 … 2 ] n m×n be a matrix. Then the column space of A, Col(A) is defined as Col(A) = Span{c , 1 , 2, c } n 17.2.1 Theorem – Suppose where A = [c c … c 1 2 n m×n is an m by n. If T(x) = Ax, then Range(T) = Col(A). Also, Col(A) = {b : Ax = b has a solution} Proof : We have shown in the previous lecture (16.1.3) that Range(T) = Span{ c , c , 1 2 …, c n} = Col(A). Furthermore, b belongs to Span{c , c ,1…, 2 } if and only if b is a linear combination of {c ,1c ,2…, c } nf and only if Ax = b has a solution. 17.2.2 Theorem – Suppose where A = [c c … c 1 2 n m×n is an m by n matrix with Rank(A) = m. Then Col(A) = ℝ . m Proof : We are given that Rank(A ) = m. Then A has a leading-1 in each row, and so m×n RREF m has no row of zeroes. Then Ax = b has a solution for all b in ℝ . That is b is linear combination of { c 1 c 2 …, c }nfor all b. We conclude that Span{c , c1, 2, c } = n m m ℝ . That is Col(A) = ℝ . 17.2.3 Example – If A is the matrix 2 Then Col(A) = ℝ since A has rank equal to 2. 17.3 Definition – Let A be an m by n matrix with row vectors {r , r , …, r }. Then m×n 1 2 m the row space of A, Row(A), is defined as Row(A) = Span{r , 1 , 2, r } m 17.3.1 Remarks – It follows immediately from the definition that the row space of m A m×n is a subspace of ℝ . T T Also note that given a matrix A m×n, Row(A) = Col(A ) and Row
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