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Section 6.1: Introduction to Linear Transformations Math 2020: Linear Algebra Ron Larson

Instructor: Dr. Vega

Math 2020: Linear Algebra

Section 6.1

Objectives

Find the image and preimage of a function. Show that a function is a linear transformation, and find a linear transformation.

Math 2020: Linear Algebra

Section 6.1

Images and Preimages of Functions.

In this chapter we will study functions between vector spaces. We usually use the notation T :V →W to denote a function between the vector spaces V and W . V is called the domain of T and W the codomain of T . If v~ ∈ V and ~ ∈ W such that T (~v ) = w w ~ , then we say that w~ is the image of ~v under T . The set of all images of vectors in V is the range of T , and the set of all v~ ∈ V sich that T (~ v) = w ~ is the preimage of w ~.

Math 2020: Linear Algebra

Section 6.1

Example Example Define the function T : R2 ! R2 such that T (v1 , v2 ) = (3v1  v2 , v1 + v2 ).

1

v = (1, 1). Find the image of ~ To do this, we just need to evaluate T (1, 1).

2

w = (2, 6). Find the preimage of ~ We need to find the vector(s) v~ = (v1 , v2 ) 2 R2 such that T (~ v ) = (2, 6).

T (1, 1) = (3(1)  (1), 1 + (1)) = (4, 0).

T (v1 , v2 ) = (2, 6) () (3v1  v2 , v1 + v2 ) = (2, 6) ()

( 3v1  v2 = 2 v1 + v2 = 6

.

solving this system, we get v1 = 1 and v2 = 5. So the preimage of (2, 6) is the set in R2 containing only the vector (4, 5).

Math 2020: Linear Algebra

Section 6.1

Linear Transformations Definition Let V and W be two vector spaces. The function T :V →W is a linear transformation of V into W if the following two properties are true for all ~u and ~v in V and any scalar c: 1) T (~ u + ~v ) = T (~u ) + T (~v ) 2) T (c ~u ) = cT (~ u) A linear transformation preserves the operations, because the same result occurs whether we perform the operations of addition and scalar multiplication before or after applying the linear transformation. Notice that the operations on the left side of properties (1) and (2) are performed in V , while the operations on the right side are performed in W. Math 2020: Linear Algebra

Section 6.1

Linear Transformation: Example Example Show that the function T : R2 ! R2 defined by T (v1 , v2 ) = (v1  v2 , v1 + 2v2 ) is a linear transformation. 1

Let ~ u = (u1 , u2 ) and ~ v = (v1 , v2 ). Then T (~ u +~ v ) = T (u1 + v1 , u2 + v2 ) = (u1 + v1  (u2 + v2 ), u1 + v1 + 2(u2 + v2 )) = ((u1  u2 ) + (v1  v2 ), (u1 + 2u2 ) + (v1 + 2v2 )) = (u1  u2 , u1 + 2u2 ) + (v1  v2 , v1 + 2v2 ) = T (~u) + T (~ v)

2

Let c 2 R and u ~ = (u1 , u2 ). Then T (c~ u ) = T (cu1 , cu2 ) = (cu1  cu2 , cu1 + 2cu2 ) = c(u1  u2 , u1 + 2u2 ) = cT (~ u)

Then, T is a linear transformation.

Math 2020: Linear Algebra

Section 6.1

Example: Functions that are NOT Linear Transformations

Example 1

Consider f : R ! R such that f (x ) = cos x . This function is NOT a linear transformation because, in general f (x1 + x2 ) = cos(x1 + x2 ) 6= cos(x1 ) + cos(x2 ) = f (x1 ) + f (x2 ).

2

f (x ) = x 3 is NOT a linear transformation from R into R since, in general 3 3 3 f (x1 + x2 ) = (x1 + x2 ) 6= x1 + x2 = f (x1 ) + f (x2 ).

3

T (v1 , v2 ) = (v1 + 1, v1 + v2 ) is NOT a linear transformation from R2 into R2 since T ((v1 , v2 ) + (u1 , u2 )) = T (v1 + u1 , v2 + u2 ) = (v1 + u1 + 1, v1 + u1 + v2 + u2 ) but T (v1 , v2 ) + T (u1 , u2 ) = (v1 + 1, v1 + v2 ) + (u1 + 1, u1 + u2 ) = (v1 + u1 + 2, v1 + v2 + u1 + u2 ). Then T (~v + ~ u ) 6= T (~ v ) + T (~u).

Math 2020: Linear Algebra

Section 6.1

Properties of Linear Transformations There are two simple linear transformations: 1

2

The Zero Transformation T : V → W defined by T (~v) = ~0 for all v~ ∈ V . The Identity Transformation T : V → V defined by T (~v) = v~ for all ~v ∈ V .

Theorem Let T be a linear transformation from V into W , and let v, ~~ u ∈ V . The properties below are true. ~V ) = 0~ . 1 T (0 W

2

T (−~ v ) = −T (~ v ).

3

~ = T (~v) − T (~ T (~u − v) u ).

4

vn , then If v~ = c1 ~ v1 + c2~ v2 + · · · + cn ~ T (~v ) = c1 T (~v1 ) + c2 T (~ v2 ) + · · · + cn T (~ vn ). Math 2020: Linear Algebra

Section 6.1

Example: Linear Transformations and Bases

Example Let T : R2 → R2 be a linear transformation such that T (1, 0) = (−1, 1), T (0, 1) = (2, 3). Find T (4, −2). Since (4, −2) = 4(1, 0) − 2(0, 1), using property (4) of the theorem: T (4, −2) = 4T (1, 0) − 2T (0, 1) = 4(−1, 1) − 2(2, 3) = (−8, −2).

Math 2020: Linear Algebra

Section 6.1

Using Matrices to define Linear Transformations In the next example we will see that matrices can be used to define linear transformations. Example Define the function T : R2 ! R3 as 0 3 v =@ 2 T (~ v ) = A.~ 1 1

v ) for ~ v = (1, 2). Using the definition of T we get Find T (~ 0 3 T (1, 2) = @ 2 1

2

1 ✓ ◆ 0 v1 1 A. . v2 2

0 1 1 ✓ ◆ 3 0 1 1 A . 2 = @0A 3 2

Show that T is a linear transformation from R2 into R3 . To do this, we need to show the two properties that define a linear transformation, namely T (~ v1 + ~v2 ) = T (~ v1 ) + T (~v2 ) and T (c~ v ) = cT (~ v ). Try this, We will do it during our lecture.

Math 2020: Linear Algebra

Section 6.1

Linear Transformations Given by a Matrix

The following theorem generalizes the previous example. Theorem Let A be an m × n matrix. The function T defined by v T (~v ) = A~ is a linear transformation from Rn into Rm . In order to conform to matrix multiplication with an m × n matrix, n × 1 matrices represent vectors in Rn and m × 1 matrices represent vectors in Rm .

Math 2020: Linear Algebra

Section 6.1

Exercises

1

Consider T : Rn → Rm defined by T (~ v ) = A~v . Find the dimensions of Rn and Rm for the following cases.

1

2

2

0

1 A =@ 0 −1 ✓ 1 3 A= 0 2

1 3 0 2 1A −2 1 ◆ 0 5 1 −1

Let T : Mm×n → Mm×n such that T (A) = ATr , where ATr represents the transpose of the matrix A. Is T a linear transformation?

Math 2020: Linear Algebra

Section 6.1

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Math 2020: Linear Algebra

Section 6.1...