Chapter 4 linear transformation

Linear transformationPhan Thi Khanh Van E-mail: khanhvanphan@hcmut.edu.vn May 13, 2021... Table of contents1 Linear transformation 2 The transformation matrix 3 Change of basis matrix an

Linear transformation

Phan Thi Khanh Van

E-mail: khanhvanphan@hcmut.edu.vn

May 13, 2021

Table of contents

1 Linear transformation

2 The transformation matrix

3 Change of basis matrix and transformation matrix

4 Kernel and Image of a linear transformation

c) We have that E = {(1; 1; 3), (1; −1; 0), (2; 0; 0)}is a basis of R3



6 101 6



b) Reflection in the liney = −3x.

c) Shear transformation with thefixed line x −axis:

S (x , y ) = (x + 3y , y )

The matrix of coordinates of the triangle: [ ~OA ~OB ~OC ] =0 2 4



f (j ) =− sin(π

3)cos(π3)

√ 3 2

1 2

2 1 −3

√ 3

2 2 −√3

1 2

√

3 +32 2√3 + 1

#

Reflection about the line y = −3x.

Choose the directional vector: e1 = (1, −3), and the normal vector

4 5

5

6

5 −45



Example 4

The projection f in Oxyz with the dot product onto the plane

P : x − y + 2z = 0is a linear transformation Findf (x , y , z)

The equationx − y + 2z = 0 has the general solution:

The transformation matrix

Letf : U → V be a linear transformation Let E = {e1, e2, , en}and

F = {f1, f2, , fm} be bases of U andV, respectively

An m × nmatrix whose thei-th column is the coordinate vector off (ei)

with respect to the basis F is called the transformation matrix off withrespect toE , F Denote: AEF =[f (e1)]F [f (e2)]F [f (en)]F

9 2

5 2



LetE andF be bases of U andV, respectively

For any linear transformation f : U → V, there exists a unique matrix

AEF satisfying: ∀x ∈ U : [f (x)]F = AEF[x ]E

For anym × n matrixAEF, there exists a unique linear transformation

∀x ∈ U : [f (x)]F = AEF[x ]E

Conclusion: For any linear transformation f, we can find 1and only 1

matrixA : f (v ) = Av If f is invertible, then f−1 has the matrixA−1 Theproduct of 2 transformationsf1 : f1(v ) = A1v andf2 : f2(v ) = A2v

corresponds toA1A2 This is where the matrix multiplication came from!



Letf be a linear transformation

f : R3→ R3 : f (x1, x2, x3) = (2x1+ x2+ x3, x1− x2− x3, 4x1− x2− x3).Find the matrix off in E = {(1, 1, 1), (1, 1, 0), (1, 0, 0)},

Change of basis matrix and transformation matrix

Letf : U → V be a linear transformation; E andE0 be 2 bases of U; F

change of basis matrix: PE0 →E = E−1E0

Then, the transformation matrix off with respect to E0:

AE0 = PE−10 →EAEPE0 →E

(AE0 and AE are 2 similar matrices)

Kernel and Image of a linear transformation

Letf be a linear transformationf : U → V

Kernel of the linear transformationf Kerf = {x ∈ U|f (x ) = 0}

Image of the linear transformation f Imf = {y ∈ V |∃x ∈ U : y = f (x )}

Letf : U → V be a linear transformation

IfU = span{e1, e2, en}, thenImf = span{f (e1), f (e2) f (en)}

Letf be a linear transformation

f : R3→ R3 : f (x1, x2, x3) = (2x1+ x2+ x3, x1− x2− x3, 4x1− x2− x3).Find the dimensions, bases ofKerf , Imf

Find a basis of Kerf:

One basis forKerf : {(0, −1, 1)}, dim(Kerf ) = 1

Find a basis of Imf:

Givenf : R3 → R3: f (1, 1, 1) = (1, 2, 1), f (1, 1, 2) = (2, 1, −1), f (1, 2, 1) =(5, 4, −1) Find the dimensions, bases of Kerf , Imf

We have that E = {e1= (1, 1, 1), e2 = (1, 1, 2), e3 = (1, 2, 1)} is a basis of

Find a basis of Kerf:

We have: x ∈ Kerf ⇔ [f (x )] = A[x ] = 0 ⇔ f (E )E−1[x ] = 0

3 The rotation f in Oxyz with respect to z-axis by 45o counterclockwise

is a linear transformation Find the dimensions, bases of Imf , Kerf

3x1− 4x2+ 3x3

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 Continue!!!

Thank you for your attention!