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LECTURE 13: SURJECTIVE LINEAR TRANSFORMATIONS

MATH 60 SPRING 2011

1. Surjective Transformations

Definition. A linear transformation T : Rn Rm is called surjective (also knownas onto) if ran T = Rm.

In other words, T is onto if each point in the codomain (i.e., the target set) Rm

is actually hit by T. More precisely, T is onto if for each y Rm there existsx Rn such that T(x) = y. In symbols this condition is:

(y Rm)(x Rn)(T(x) = y ).

Recall that the symbol is read for every or for each. The importance of sur-jective linear transformations is their relationship with systems of linear equations.In particular, the concepts of surjectivity and solvability are closely related:

Theorem 1. If A = (v1|v2| |vn) is an m n matrix, then the following areequivalent

(i) TA : Rn Rm is surjective,

(ii) span{v1, v2, , vn} = Rm,

(iii) for each y Rm the systemAx = y is consistent.

The span span{v1,v2, ,vn} of the columns ofA is called the columnspace ofA, denoted colspace(A). With this terminology, condition (ii) can be phrased ascolspace(A) = Rm.

In the case where A is an n n matrix (i.e., when TA : Rn Rn is a linearoperator on Rn), we can say significantly more:

Theorem 2 (Master Theorem for n n Matrices). If A = (v1|v2| |vn) is ann n matrix and ifTA : R

n Rn denotes the associated linear operator

TA(x) = Ax,

then the following statements are equivalent:

(i) A is invertible,

(ii) Ax = 0 has only the trivial solution (i.e. x = 0),

(iii) The reduced row echelon form ofA is In,(iv) A is expressible as the product of elementary matrices,

(v) Ax = y is consistent for each y Rn,

(vi) Ax = y has exactly one solution for each y Rn,

(vii) detA = 0,

Date: Wednesday 2/16/11.

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2 MATH 60 SPRING 2011

(viii) TA is surjective (i.e., onto),

(ix) span{v1, v2, , vn} = Rn.

Example 1. This example indicates some differences between square matrices andnon-square matrices. In particular, it helps illustrate how the Master Theorem forn n matrices does not apply to non-square matrices. Since the matrix

A =

1 0 00 1 0

is 2 3, it gives rise to a linear transformation TA : R3 R2. Let us examine how

the statements in the Master Theorem relate to A:

(i) Meaningless since A is not square,

(ii) False, since x = (0, 0, c) is also a solution for any value of c,

(iii) False, since A is not square,

(iv) False, since elementary matrices are square,

(v) True. Given y = (a, b), let x = (a,b,c) where c is arbitrary,

(vi) False. In the preceding statement, c is arbitrary,

(vii) Meaningless since A is not square,

(viii) True. Restatement of (v),

(ix) True. Restatement of (viii).

As this example shows, the Master Theorem for n n matrices does not applyto non-square matrices.