Chapter 4 Linear Transformations

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Chapter 4 Linear Transformations

• 4.1 Introduction to Linear Transformations
• 4.2 The Kernel and Range of a Linear
  Transformation
• 4.3 Matrices for Linear Transformations
• 4.4 Transition Matrices and Similarity

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4.1 Introduction to Linear Transformations

• A linear transformation is a function T that maps
  a vector space V into another vector space W

V the domain of T W the codomain of T
Two axioms of linear transformations
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• Image of v under T

If v is in V and w is in W such that
Then w is called the image of v under T .

• the range of T
• The set of all images of vectors in V.

• the preimage of w
• The set of all v in V such that T(v)w.

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• Notes

(1) A linear transformation is said to be
operation preserving.
(2) A linear transformation
from a vector space into itself is called a
linear operator.
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• Ex (Verifying a linear transformation T from R2
  into R2)

Pf
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Therefore, T is a linear transformation.
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• Ex (Functions that are not linear
  transformations)

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• Notes Two uses of the term linear.

(1) is called a linear
function because its graph is a line. But
(2) is not a linear
transformation from a vector space R into R
because it preserves neither vector addition nor
scalar multiplication.
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• Zero transformation

• Identity transformation

• Thm 4.1 (Properties of linear transformations)

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• Ex (Linear transformations and bases)

Let be a linear
transformation such that
Find T(2, 3, -2).
Sol
(T is a L.T.)
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• Ex (A linear transformation defined by a matrix)

The function is defined as
Sol
(vector addition)
(scalar multiplication)
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• Thm 4.2 (The linear transformation given by a
  matrix)

Let A be an m?n matrix. The function T defined by
is a linear transformation from Rn into Rm.

• Note

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• Rotation in the plane

Show that the L.T. given by
the matrix
has the property that it rotates every vector in
R2 counterclockwise about the origin through the
angle ?.
Sol
(polar coordinates)
r the length of v ?the angle from the positive
x-axis counterclockwise to the vector v
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rthe length of T(v) ? ?the angle from
the positive x-axis counterclockwise to
the vector T(v)
Thus, T(v) is the vector that results from
rotating the vector v counterclockwise through
the angle ?.
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• A projection in R3

The linear transformation
is given by
is called a projection in R3.
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• A linear transformation from Mm?n into Mn ?m

Show that T is a linear transformation.
Sol
Therefore, T is a linear transformation from Mm?n
into Mn ?m.
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4.2 The Kernel and Range of a Linear
Transformation

• Kernel of a linear transformation T

Let be a linear transformation
Then the set of all vectors v in V that satisfy
is called the kernel of T and is
denoted by ker(T).
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• Ex 2 The kernel of the zero and identity
  transformations

(a) T(v)0 (the zero transformation
)
(b) T(v)v (the identity transformation
)
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• Finding the kernel of a linear transformation

Sol
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• Thm 4.3 (The kernel is a subspace of V)

The kernel of a linear transformation
is a subspace of the domain V.
Pf

• Note
• The kernel of T is also called the nullspace of
  T.

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Finding a basis for the kernel
Find a basis for ker(T) as a subspace of R5.
Sol
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• Corollary to Thm 4.3

• Thm 4.4 The range of T is a subspace of W

Pf
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• Notes

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Finding a basis for the range of a linear
transformation
Find a basis for the range of T.
Sol
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• Rank of a linear transformation T V?W

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• Thm 4.5 Sum of rank and nullity

Pf
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• Finding the rank and nullity of a linear
  transformation

Sol
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• One-to-one

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• Onto

i.e., T is onto W when range(T)W.
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• Thm 4.6 (One-to-one linear transformation)

Pf
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• One-to-one and not one-to-one linear
  transformation

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• Thm 4.7 (Onto linear transformation)

• Thm 4.8 (One-to-one and onto linear
  transformation)

Pf
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• Ex

Sol
TRn?Rm dim(domain of T) rank(T) nullity(T) 1-1 onto
(a)TR3?R3 3 3 0 Yes Yes
(b)TR2?R3 2 2 0 Yes No
(c)TR3?R2 3 2 1 No Yes
(d)TR3?R3 3 2 1 No No
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• Isomorphism

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• Ex (Isomorphic vector spaces)

The following vector spaces are isomorphic to
each other.
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4.3 Matrices for Linear Transformations

• Two representations of the linear transformation
  TR3?R3

• Three reasons for matrix representation of a
  linear transformation

• It is simpler to write.
• It is simpler to read.
• It is more easily adapted for computer use.

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• Thm 4.10 (Standard matrix for a linear
  transformation)

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• Pf

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• Ex (Finding the standard matrix of a linear
  transformation)

Sol
Vector Notation Matrix
Notation
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• Composition of T1 Rn?Rm with T2 Rm?Rp

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• Pf

• But note

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• Ex (The standard matrix of a composition)

Sol
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• Inverse linear transformation

• Note
• If the transformation T is invertible, then the
  inverse is unique and denoted by T1 .

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• Existence of an inverse transformation

1. T is invertible.
2. T is an isomorphism.
3. A is invertible.

• Note
• If T is invertible with standard matrix A, then
  the standard matrix for T1 is A1 .

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• Ex (Finding the inverse of a linear
  transformation)

Show that T is invertible, and find its inverse.
Sol
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• the matrix of T relative to the bases B and B'

Thus, the matrix of T relative to the bases B and
B' is
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• Transformation matrix for nonstandard bases

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• Ex (Finding a transformation matrix relative to
  nonstandard bases)

Sol
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• Check

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• Notes

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4.4 Transition Matrices and Similarity
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• Two ways to get from to

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• Ex

Sol
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with
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• Similar matrix
• For square matrices A and A of order n, A is
  said to be similar to A if there exist an
  invertible matrix P such that

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• Ex (A comparison of two matrices for a linear
  transformation)

Sol
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