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Linear Algebra

CHAPTER 4. Linear transformation and Inner

Product

Lecturer: Dr. Nguyen Nhu Ngoc

Vietnam National University - HCMC International University

2025-

CONTENTS

(^1) Matrix as linear transformation

2 Orthogonal basis

(^3) Gram-Schmidt Process

Linear transformation

Transformation by Matrix Multiplication

Example

Consider the matrix A =

. By matrix multiplication, A

transforms vectors in R^3 into vectors in R^2.

Consider the vector

x y z

. Transforming this vector by A looks like:

x y z

x + 2 y 2 x + y

For example:  1 2 0 2 1 0

Transformations

Definition

A transformation is a function T : R n^ → R m, sometimes written

R n^ −→T R m,

and is called a transformation from R n^ to R m. If m = n, then we say T is a transformation of R n.

What do we mean by a function?

Informally, a function T : R n^ → R m^ is a rule that assigns exactly one vector of R m^ to each vector of R n. We use the notation T(⃗ x) to mean the transformation T applied to the vector ⃗x.

Definition

If T acts by matrix multiplication of a matrix A (such as the previous example), we call T a matrix transformation, and write TA(⃗ x) = A⃗ x.

Specifying the Action of a Transformation

Example

T : R^3 → R^4 defined by

T

a b c

a + b b + c a − c c − b

is a transformation that transforms the vector

 (^) in R^3 into the vector

T

Properties of Linear Transformations

Let T : R n^ → R m^ be a linear transformation, and let ⃗x ∈ R n. Since T preserves scalar multiplication, (^1) T( 0 ⃗ x) = 0 T(⃗ x) implying T(⃗ 0 ) = ⃗ 0 , so T preserves the zero vector. (^2) T((− 1 )⃗ x) = (− 1 )T(⃗ x), implying T(−⃗ x) = −T(⃗ x), so T preserves the negative of a vector.

Suppose ⃗x 1 , ⃗x 2 ,... , ⃗xk are vectors in R n^ and

⃗ y = a 1 ⃗ x 1 + a 2 ⃗ x 2 + · · · + ak⃗ xk

for some a 1 , a 2 ,... , ak ∈ R. Then (^3) T(⃗ y) = T(a 1 ⃗ x 1 + a 2 ⃗ x 2 + · · · + ak⃗ xk) = a 1 T(⃗ x 1 ) + a 2 T(⃗ x 2 ) + · · · + ak T(⃗ xk),

i.e., T preserves linear combinations.

Problem

Let T : R^3 → R^4 be a linear transformation such that

T

 and^ T

.^ Find^ T

Solution

The only way it is possible to solve this problem is if

 

 (^) is a linear combination of

 (^) and

i.e., if there exist a, b ∈ R so that  

 (^) = a

 (^) + b

Solution (continued)

We now use that fact that linear transformations preserve linear combinations, implying that

T

 = T

= T

 − 2 T

 −^2

Therefore, T

Problem

Let T : R^4 → R^3 be a linear transformation such that

T

 (^) and T

 (^). Find T

Final Answer

T

Some Special Matrix Transformations

Example (The Zero Transformation)

If A is the m × n matrix of zeros, then the transformation T : R n^ → R m induced by A is called the zero transformation because for every vector ⃗x in R n T(⃗ x) = A⃗ x = 0 ⃗ x = ⃗0.

Note that the first zero is the matrix A, while the second zero is the zero vector of R m. The zero transformation is usually written as T = 0.

Example (The Identity Transformation)

The transformation of R n^ induced by In, the n × n identity matrix, is called the identity transformation because for every vector ⃗x in R n,

T(⃗ x) = In⃗ x = ⃗x.

The identity transformation on R n^ is usually written as (^1) R n^.

Example (Revisited)

Recall T : R^3 → R^4 defined by

T

a b c

a + b b + c a − c c − b

Is T a matrix transformation?

Consider A =

, then

A

a b c

a b c

a + b b + c a − c c − b

 =^ T

a b c

So in this case T is a matrix transformation!

Example (continued)

We have T : R^2 → R^2 defined by

T(⃗ x) = ⃗x +

for all ⃗x ∈ R^2.

Since every matrix transformation is a linear transformation, we consider T(⃗ 0 ), where ⃗ 0 is the zero vector of R^2.

T

violating one of the properties of a linear transformation. Therefore, T is not a linear transformation, and hence is not a matrix transformation. Can you see any other reasons why T is not a matrix transformation?

Matrix Transformations

Theorem

Let T : R n^ → R m^ be a linear transformation. Then we can find an n × m matrix A such that T(⃗ x) = A⃗ x

In this case, we say that T is induced, or determined, by A and we write

TA(⃗ x) = A⃗ x

Corollary

A transformation T : R n^ → R m^ is a linear transformation if and only if it is a matrix transformation.