The Matrix Representation of a Linear Transformationn

THE MATRIX REPRESENTATION OF A LINEAR TRANSFORMATION

Definition: Ordered Basis :

Let V be a finite-dimensional vector space. An ordered basis for V is a basis for V endowed with a specific order; that is, an ordered basis for V is a finite sequence of linearly independent vectors in V that generates V.

Example

If F³, α = {e₁, e₂, e₃} can be considered an ordered basis. Also {e₂, e₁, e₃} is an ordered basis, but α ≠ β as ordered bases.

Note: For the vector space Fⁿ, we call (e₁, e₂, ..., e_n} the standard ordered basis for Fⁿ.

Similarly, for the vector space P_n(F), we call {1, x, ..., xⁿ} the standard ordered basis for P_n(F)

Definition :

Let  be an ordered basis for a finite-dimensional vector space V. For  be the unique scalars such that

We define the coordinate vector of x relative to β, denoted [x]_β by

Example

Note :

The promised matrix representation of a linear transformation. Suppose that V and W are finite-dimensional vector spaces with ordered bases  respectively. Let T : V → W be linear. Then for each j, 1 ≤ j ≤ n, there exist unique scalars  such that

Definition :

Using the notation above, we call the m x n matrix A defined by A_ij = a_ij the matrix representation of T in the ordered basis β and  then we write 

Note: The j^th column of A is simply  Also observe that If U : V → W is a linear transformation such that  then U = T

Example 1.

Note: Plural of basis is bases

Example 2

Problem 1

Solution :

Similarly, for another basis,

Problem 2

Solution :

Similarly, for another two basis.

Problem 3

Solution :

Problem 4

Solution :

Problem 5

Solution :

First fine the images of the basis vectors

Write these images as a linear combination of the basis vectors in γ

Solve we get

Hence, the vector (x,y,z) can be written as a linear combination of the basis vectors in

Thus,

The vector T(0,1) = (-1,0,1) can be written as follows:

Hence, the matrix of the linear transformation with respect to the bases β ans γ is

Problem 6

Solution :

Problem 7

Solution :

(b) T : V → Fⁿ, T : V → V

Problem 1

Let V be an n-dimensional vector space with an ordered basis β. Define T : V → Fⁿ by T(x) = [x]_β. Prove that T is linear.

Solution :

Given that V is an n dimensional vector space over the field F and β = {v₁, ..., v_n} is its basis.

So, every vector in V can be written as a linear combination of β = {v₁, ..., v_n}

On the other hand, T : V → Fⁿ is a function such that T(x) = [x]_β

Note that [x]_β is the column of the coefficients of the linear combination that x is written using β = {v₁, ... v_n}

Now, by the properties of matrices, we know that

Problem 2

Solution :

A complex number z = a + ib can be written as (a, b)

But both a and b are real numbers.

.'. the set of complex numbers C can be followed as R²

To prove T is linear,

.'. T is a linear transformation.

Given basis is {1, i}

i.e., {1+Oi, 0+ li} or {(1, 0), (0, 1)}

(i) We write the images of the basis under T.

(ii) We write the images as the linear combinations of the co-domain basis.

While the co-domain is also C, we get

(iii) We write the coefficients of the linear combinations as the columns of the matrix.

(c) T, U : V → W, T : V → W

Definition :

Let T, U : V → W be arbitrary functions, where V and W are vector spaces over F, and let a Є F.

THEOREM 1.

Let V and W be vector spaces over a field F, and let T, U : V → W be linear.

(a) For all a Є F, a T + U is linear.

(b) Using the operations of addition and scalar multiplication in the preceding definition, the collection of all linear transformations from V to W is a vector space over F.

Proof :

So aT + U is linear.

(b) Let V and W be two vector spaces over F : T, U : V → W be linear transformation.

Use the operations of addition and scalar multiplication to prove that the collection of all linear transformation of this type from V to W is a vector space over F.

Suppose a Є F, then, define T + U : V → W by,

Clearly, the transformation T + U is linear.

Assume S is the set of all linear transformation as defined above.

For each pair of transformations T, U in S there is a unique transformation T+U in S defined by (T+U)(x) = T(x) + U(x)

Also, for all a F and each transformation T in S there is a unique transformation a T in S

i.e., defined by (a T)(x) = a T(x)

(i) Commulativity of addition: To prove that for all T, U Є S. T + U = U + T.

Take T + U, which is defined as,

(ii) Associativity of addition: Prove that for all T, U, P Є S.

i.e., To prove (T + U) + P = T + (U + P)

Therefore, for all T, U, P in S,

(T + U) + P = T + (U + P)

(iii) Additive Identity: To prove that there exists a transformation in S, denoted by T₀, that is defined as T₀(x) = 0, such that T + T₀ = T for each T in S.

Here, T₀ acts as zero transformation and play the role of zero vectors.

(iv) Additive inverse: To prove that for each T in S, there exists a -T in S such that T + (-T) = 0

From the definition (2), for -1 Є F and for each T in S, there is a unique transformation -1. T in S defined by

Therefore, for each T in S, there exists a -T in S such that T + (-T) = 0

(v) Scalar identity: To prove that for each T is S, 1. T = T

From the definition (2), for 1 Є F and for each T in S, there is a unique transformation 1. T in S defined by,

Therefore, for each T in S, 1.T = T

(vi) Scalar Associativity: To prove that for each pair of elements a, b in F, and for each T in S.

Hence, from the above proved conditions, conclude that S is a vector space.

That is, the set of all linear transformations defined from the vector space V to the vector space W is a vector space over F.

Definitions :

Let V and W be vector spaces over F.

We denote the vector space of all linear transformations from V into W by L(V, W).

In the case that V = W, we write L(V) instead of L(V, W).

THEOREM 2.

Let V and W be finite-dimensional vector spaces with ordered bases β and γ, respectively, and let T, U : V → W be linear transformations. Then

Proof :

(b) If T : V → W is a line transformation, then the matrix representation of T is obtained in the following manner.

(i) β = {v₁, ..., v_n} is the basis of V and we find the images of β = {v₁, ..., v_n} under T.

(ii) We write the images as the linear combinations of γ = {w₁, ..., w_m}

(iii) We write the coefficients these linear combinations as the columns of a matrix

Let β and γ be the standard ordered bases of R² and R³, respectively. Then

If we compute T + U using the preceding definitions, we obtain

Problem 1

Let V and W be vector spaces, and let T and U be non-zero linear transformations from V into W. If R(T) ∩ R(U) = {0}, prove that {T, U} is a linearly independent subset of L(V, W).

Solution :

Let T and U be non-zero linear transformations from V into the vector space W.

R(T) ∩ R(U) = {0}

To prove that T and U are linearly independent

We consider  is the zero transformation.

Applying any vector x in V on both sides, we get (aT + b U) (x) = 0 (x)

By the definition of addition and scalar multiplication of transformation, we get

So, the vectors ax and -bx have the same image under T and U.

In other words, ax and -bx are the common range of T and U.

But by hypothesis,

We have R(T) ∩ R(U) = {0}

=> ax = 0 and -bx = 0 for an arbitrary vector x in V.

Consequently, there is the only possibility that a = 0 and b = 0

Thus, we have prove that (a T + b U) (x) =  => a = 0 and b = 0

Therefore, T and U are linearly independent.

Problem 2.

Let V and W be vector spaces, and let S be a subset of V. Define S⁰ = {T Є L (V, W) : T(x) = 0 for all x Є S}. Prove the following statement.

S⁰ is a subspace of L (V, W)

Solution :

Problem 3

Solution :

EXERCISE 5.2

1. Label the following statements as true or false. Assume that V and W are finite-dimensional vector spaces with ordered basis β and γ, respectively, and T, U : V → W are linear transformations.

(i) For any scalar a, a T + U is a linear transformation from V to W.

(iii) L(V, W) is a vector space.

2. Let β and γ be the standard ordered bases for Rⁿ and R^m respectively. For each linear transformation T : Rⁿ → R^m, compute  is defined by