How To Find The Matrix Of A Linear Transformation

The matrix of a linear transformation provides a powerful way to represent and perform linear operations in a structured and computationally efficient manner. It acts as a bridge, connecting abstract linear transformations with the concrete world of matrices and vectors, enabling us to analyze and manipulate them using the tools of linear algebra. Finding this matrix representation is a fundamental skill in linear algebra, allowing for efficient computation, easier analysis, and a deeper understanding of linear transformations themselves.

Understanding Linear Transformations

A linear transformation is a function T that maps vectors from one vector space V to another vector space W, while preserving vector addition and scalar multiplication. More formally, for any vectors u, v in V and any scalar c, the following two properties must hold:

• T(u + v) = T(u) + T(v)
• T(cu*) = cT(u)

Examples of linear transformations include rotations, reflections, scaling, and projections. These transformations are ubiquitous in mathematics, physics, computer graphics, and many other fields.

Why Find the Matrix of a Linear Transformation?

Representing a linear transformation as a matrix offers several advantages:

• Computational Efficiency: Once you have the matrix, applying the transformation to a vector simply becomes a matrix-vector multiplication, a well-defined and computationally efficient operation.
• Composition of Transformations: The composition of two linear transformations can be represented by multiplying their corresponding matrices.
• Analysis and Properties: The matrix representation allows us to analyze properties of the transformation, such as its rank, nullity, eigenvalues, and eigenvectors, using matrix algebra techniques.
• Change of Basis: The matrix representation changes when the basis of the vector spaces changes, allowing us to understand how the transformation behaves under different coordinate systems.
• Generalization: Working with matrices allows us to generalize linear transformations to higher-dimensional spaces more easily.

The Fundamental Theorem

The process of finding the matrix of a linear transformation relies on a fundamental theorem in linear algebra:

Theorem: Let V and W be finite-dimensional vector spaces, and let {v₁, v₂, ..., vₙ} be a basis for V. Then a linear transformation T: V → W is completely determined by the images of the basis vectors, T(v₁), T(v₂), ..., T(vₙ).

This theorem tells us that to fully understand a linear transformation, we only need to know what it does to a basis of the input vector space. This is the key to finding the matrix representation.

Steps to Find the Matrix of a Linear Transformation

Here's a step-by-step guide to finding the matrix of a linear transformation T: V → W with respect to given bases for V and W:

1. Choose Bases for V and W:

• Select a basis B = {v₁, v₂, ..., vₙ} for the input vector space V. This basis is crucial because we will find the images of these basis vectors under the transformation. The choice of basis can affect the simplicity of the resulting matrix, so consider choosing a basis that simplifies the calculations. A common choice is the standard basis (e.g., {(1, 0), (0, 1)} for R², {(1, 0, 0), (0, 1, 0), (0, 0, 1)} for R³).
• Select a basis C = {w₁, w₂, ..., wₘ} for the output vector space W. This basis will be used to express the images of the basis vectors of V as linear combinations. Again, the choice of basis can affect the simplicity of the resulting matrix.

2. Find the Images of the Basis Vectors of V:

• Apply the linear transformation T to each basis vector vᵢ in B:
  • Compute T(v₁)
  • Compute T(v₂)
  • ...
  • Compute T(vₙ)
• Each T(vᵢ) will be a vector in the output vector space W.

3. Express the Images as Linear Combinations of the Basis Vectors of W:

• For each T(vᵢ), express it as a linear combination of the basis vectors in C:
  • T(v₁) = a₁₁w₁ + a₂₁w₂ + ... + aₘ₁wₘ
  • T(v₂) = a₁₂w₁ + a₂₂w₂ + ... + aₘ₂wₘ
  • ...
  • T(vₙ) = a₁ₙw₁ + a₂ₙw₂ + ... + aₘₙwₘ
• The coefficients aᵢⱼ are scalars. Finding these coefficients is often the most computationally intensive part of the process. This often involves solving systems of linear equations.

4. Construct the Matrix:

• The matrix A of the linear transformation T with respect to the bases B and C is an m x n matrix, where the j-th column of A consists of the coefficients from the linear combination of T(vⱼ) in terms of the basis C. That is:

  A =  [ a₁₁  a₁₂  ... a₁ₙ ]
       [ a₂₁  a₂₂  ... a₂ₙ ]
       [ ...  ...  ... ... ]
       [ aₘ₁  aₘ₂  ... aₘₙ ]
  

• The entry aᵢⱼ in the matrix A represents the i-th component of T(vⱼ) with respect to the basis C.

Therefore, the matrix A represents the linear transformation T with respect to the chosen bases B and C. We often denote this as A = [T]*_B,C. If V = W and B = C, we simply write A = [T]*_B.

Example: A Linear Transformation from R² to R²

Let's consider the linear transformation T: R² → R² defined by T( x, y ) = ( x + y, 2x - y ). We will find the matrix of T with respect to the standard basis B = { e₁ = (1, 0), e₂ = (0, 1) } for both the input and output spaces.

1. Choose Bases:

• B = { (1, 0), (0, 1) } for the input R²
• B = { (1, 0), (0, 1) } for the output R²

2. Find the Images of the Basis Vectors:

• T(e₁) = T(1, 0) = (1 + 0, 2(1) - 0) = (1, 2)
• T(e₂) = T(0, 1) = (0 + 1, 2(0) - 1) = (1, -1)

3. Express the Images as Linear Combinations:

Since we are using the standard basis, this step is straightforward. The images are already expressed in terms of the standard basis:

• (1, 2) = 1(1, 0) + 2(0, 1)
• (1, -1) = 1(1, 0) + (-1)(0, 1)

4. Construct the Matrix:

The coefficients of the linear combinations form the columns of the matrix:

A = [ 1  1 ]
    [ 2 -1 ]

Therefore, the matrix of the linear transformation T with respect to the standard basis is:

A = [ 1  1 ]
    [ 2 -1 ]

To apply the transformation to a vector v = (x, y), we can simply multiply the matrix A by the vector v (expressed as a column vector):

[ 1  1 ] [ x ]  =  [ x + y ]
[ 2 -1 ] [ y ]     [ 2x - y ]

This confirms that the matrix A correctly represents the linear transformation T.

Example: A Linear Transformation from P₁(R) to R²

Let's consider the linear transformation T: P₁(R) → R² defined by T( p(t) ) = (p(0), p(1)), where P₁(R) is the vector space of polynomials of degree at most 1 with real coefficients. We will find the matrix of T with respect to the basis B = {1, t} for P₁(R) and the standard basis C = { (1, 0), (0, 1) } for R².

1. Choose Bases:

• B = {1, t} for P₁(R)
• C = { (1, 0), (0, 1) } for R²

2. Find the Images of the Basis Vectors:

• T(1) = (1, 1) (Since the polynomial is the constant 1, its value at both t=0 and t=1 is 1)
• T(t) = (0, 1) (Since the polynomial is t, its value at t=0 is 0 and its value at t=1 is 1)

3. Express the Images as Linear Combinations:

Since we are using the standard basis for R², this step is straightforward:

• (1, 1) = 1(1, 0) + 1(0, 1)
• (0, 1) = 0(1, 0) + 1(0, 1)

4. Construct the Matrix:

The coefficients of the linear combinations form the columns of the matrix:

A = [ 1  0 ]
    [ 1  1 ]

Therefore, the matrix of the linear transformation T with respect to the bases B and C is:

A = [ 1  0 ]
    [ 1  1 ]

To apply the transformation to a polynomial p(t) = a + bt, we can represent the polynomial as a coordinate vector with respect to the basis B: [p(t) ]*_B = [ a, b ]*. Then, we multiply the matrix A by this coordinate vector:

[ 1  0 ] [ a ]  =  [ a ]
[ 1  1 ] [ b ]     [ a + b ]

The resulting vector [ a, a + b ]* represents the coordinates of the image vector in R² with respect to the standard basis. Since the standard basis is { (1, 0), (0, 1) }, the image vector is simply (a, a + b). This matches the definition of T: T(a + bt) = (a, a + b), since p(0) = a and p(1) = a + b.

Dealing with Non-Standard Bases

When dealing with non-standard bases, expressing the images of the basis vectors as linear combinations requires solving a system of linear equations. Let's illustrate this with an example.

Let T: R² → R² be defined by T( x, y ) = (2x + y, x - 3y). Let B = { (1, 1), (1, -1) } be a non-standard basis for the input R², and let C = { (1, 0), (0, 1) } be the standard basis for the output R². We want to find the matrix [T]*_B,C.

1. Choose Bases:

• B = { (1, 1), (1, -1) } for the input R²
• C = { (1, 0), (0, 1) } for the output R²

2. Find the Images of the Basis Vectors:

• T(1, 1) = (2(1) + 1, 1 - 3(1)) = (3, -2)
• T(1, -1) = (2(1) + (-1), 1 - 3(-1)) = (1, 4)

3. Express the Images as Linear Combinations:

Since C is the standard basis, this step is again straightforward:

• (3, -2) = 3(1, 0) + (-2)(0, 1)
• (1, 4) = 1(1, 0) + 4(0, 1)

4. Construct the Matrix:

A = [ 3  1 ]
    [ -2 4 ]

Now, let's consider a case where both bases are non-standard. Let T: R² → R² be defined as before, T( x, y ) = (2x + y, x - 3y). Let B = { (1, 1), (1, -1) } for the input space and C = { (1, 2), (0, 1) } for the output space. We want to find [T]*_B,C.

1. Choose Bases:

• B = { (1, 1), (1, -1) } for the input R²
• C = { (1, 2), (0, 1) } for the output R²

2. Find the Images of the Basis Vectors:

As before:

• T(1, 1) = (3, -2)
• T(1, -1) = (1, 4)

3. Express the Images as Linear Combinations:

This is where things get more interesting. We need to find scalars a and b such that:

• (3, -2) = a(1, 2) + b(0, 1) and
• (1, 4) = c(1, 2) + d(0, 1)

This gives us two systems of linear equations:

• For (3, -2):

  • a = 3
  • 2a + b = -2 => 2(3) + b = -2 => b = -8

• For (1, 4):

  • c = 1
  • 2c + d = 4 => 2(1) + d = 4 => d = 2

4. Construct the Matrix:

The coefficients we found form the columns of the matrix:

A = [  3   1 ]
    [ -8   2 ]

Therefore, the matrix of the linear transformation T with respect to the bases B and C is:

A = [  3   1 ]
    [ -8   2 ]

Important Considerations

• Order Matters: The order of the basis vectors in B and C is crucial. Changing the order will change the matrix representation.
• Linear Independence: The set of basis vectors B and C must be linearly independent and span the vector spaces V and W, respectively.
• Non-Uniqueness: While the linear transformation T is unique, its matrix representation is not. It depends on the choice of bases B and C.
• Software Tools: For complex linear transformations and high-dimensional vector spaces, software tools like MATLAB, Mathematica, or Python with libraries like NumPy can be invaluable for performing the calculations.

Common Applications

Finding the matrix of a linear transformation is a fundamental skill with widespread applications:

• Computer Graphics: Transformations like rotations, scaling, and translations are represented by matrices to manipulate objects in 2D and 3D space.
• Image Processing: Linear transformations are used for image filtering, edge detection, and image compression.
• Physics: Linear transformations are used to describe rotations, reflections, and other transformations in physical systems. They are also crucial in quantum mechanics.
• Engineering: Linear transformations are used in structural analysis, circuit analysis, and control systems.
• Data Science: Linear transformations are used in dimensionality reduction techniques like Principal Component Analysis (PCA).

Conclusion

Finding the matrix of a linear transformation is a core concept in linear algebra that bridges the gap between abstract linear operations and concrete matrix representations. By following the steps outlined above, you can effectively find the matrix representation of any linear transformation with respect to chosen bases. This skill is essential for understanding, analyzing, and applying linear transformations in various fields of science, engineering, and computer science. Remember that the choice of bases influences the matrix representation, and practice is key to mastering this fundamental concept. The ability to represent linear transformations as matrices opens the door to a powerful set of tools for solving a wide range of problems.