Find The Matrix Of The Linear Transformation Calculator

This calculator helps you find the matrix of a linear transformation T: Rⁿ → R^m given the images of the standard basis vectors of Rⁿ.

Calculator

Visualization (R² to R²)

Understanding the Matrix of a Linear Transformation

What is the Matrix of a Linear Transformation?

A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. For finite-dimensional vector spaces, like Rⁿ and R^m, every linear transformation T: Rⁿ → R^m can be represented by an m × n matrix, often denoted as [T] or A. This matrix is called the matrix of the linear transformation relative to the standard bases of Rⁿ and R^m.

If v is a vector in Rⁿ, and T(v) is its image in R^m, then T(v) = Av, where A is the matrix of the transformation and the vectors are treated as column matrices. The matrix of linear transformation calculator helps you find this matrix A by knowing how T acts on the standard basis vectors of Rⁿ.

Anyone studying linear algebra, computer graphics, physics, engineering, or any field that uses vector transformations will find the concept and our matrix of linear transformation calculator useful.

A common misconception is that a linear transformation *is* a matrix. While a matrix can represent a linear transformation (with respect to chosen bases), the transformation is the underlying function, and its matrix representation can change if the bases change. Our calculator uses the standard bases.

Matrix of Linear Transformation Formula and Mathematical Explanation

Let T: Rⁿ → R^m be a linear transformation. Let {e₁, e₂, …, e_n} be the standard basis for Rⁿ (where e_i is a vector with 1 in the i-th position and 0s elsewhere), and {f₁, f₂, …, f_m} be the standard basis for R^m.

The matrix A of the linear transformation T is formed by finding the images of the basis vectors e₁, e₂, …, e_n under T and expressing these images as column vectors in terms of the basis of R^m. Specifically, the j-th column of A is the coordinate vector of T(e_j) with respect to the standard basis of R^m.

So, if T(e₁) = [a₁₁, a₂₁, …, a_m1]^T, T(e₂) = [a₁₂, a₂₂, …, a_m2]^T, …, T(e_n) = [a_1n, a_2n, …, a_mn]^T, then the matrix A is:

A = [ T(e₁) | T(e₂) | … | T(e_n) ] =

a11	a12	…	a1n
a21	a22	…	a2n
…	…	…	…
am1	am2	…	amn

This m × n matrix A is the standard matrix for the linear transformation T. Our matrix of linear transformation calculator implements this principle.

Variables Table

Variable	Meaning	Unit	Typical Range
n	Dimension of the domain (R^n)	Integer	2, 3 (in this calculator)
m	Dimension of the codomain (R^m)	Integer	2, 3 (in this calculator)
ei	i-th standard basis vector of R^n	Vector	e.g., [1, 0]^T, [0, 1]^T for R^2
T(ei)	Image of ei under T in R^m	Vector	Components are real numbers
A (or [T])	The m × n matrix of T	Matrix	m rows, n columns
aij	Element in the i-th row and j-th column of A	Real number	Any real number

Variables involved in finding the matrix of a linear transformation.

Practical Examples (Real-World Use Cases)

Example 1: Rotation in R²

Consider a linear transformation T: R² → R² that rotates vectors counterclockwise by an angle θ.

The standard basis vectors in R² are e₁ = [1, 0]^T and e₂ = [0, 1]^T.

T(e₁) = [cos(θ), sin(θ)]^T

T(e₂) = [-sin(θ), cos(θ)]^T

The matrix of T is A = [T(e₁) | T(e₂)] =

cos(θ)	-sin(θ)
sin(θ)	cos(θ)

If θ = 90°, cos(90°)=0, sin(90°)=1. T(e₁)=[0,1]^T, T(e₂)=[-1,0]^T. Matrix A = [[0, -1], [1, 0]]. Using the matrix of linear transformation calculator with n=2, m=2, T(1,0)=(0,1), T(0,1)=(-1,0) gives this matrix.

Example 2: Projection onto the xy-plane in R³

Consider a linear transformation T: R³ → R³ that projects vectors onto the xy-plane. T(x, y, z) = (x, y, 0).

Standard basis vectors in R³: e₁=[1,0,0]^T, e₂=[0,1,0]^T, e₃=[0,0,1]^T.

T(e₁) = T(1,0,0) = (1,0,0) = [1,0,0]^T

T(e₂) = T(0,1,0) = (0,1,0) = [0,1,0]^T

T(e₃) = T(0,0,1) = (0,0,0) = [0,0,0]^T

The matrix of T is A = [T(e₁)|T(e₂)|T(e₃)] =

1	0	0
0	1	0
0	0	0

Using the matrix of linear transformation calculator with n=3, m=3 and inputs T(1,0,0)=(1,0,0), T(0,1,0)=(0,1,0), T(0,0,1)=(0,0,0) will yield this matrix.

How to Use This Matrix of Linear Transformation Calculator

1. Select Dimensions: Choose the dimension of the domain (n) and the codomain (m) from the dropdown menus. The calculator supports n, m = 2 or 3.
2. Enter Image Vectors: Based on your selected dimensions, input fields will appear for the components of the images of the standard basis vectors T(e₁), T(e₂), … T(e_n). For example, if n=2, m=2, enter the components of T(1,0) and T(0,1).
3. Calculate: Click the “Calculate Matrix” button, or the matrix will update automatically as you type if you’ve already clicked it once.
4. View Results: The resulting m × n matrix A will be displayed in the “Matrix of the Linear Transformation (A)” section. The “Images of Basis Vectors” section will confirm the vectors T(e_i) you entered or calculated from inputs.
5. Visualization (for R² to R²): If you select n=2 and m=2, a graph will show the standard basis vectors and their images under the transformation.
6. Reset/Copy: Use “Reset” to clear inputs and “Copy Results” to copy the matrix and intermediate values.

The matrix of linear transformation calculator provides the standard matrix, which is essential for applying the transformation to any vector in the domain.

Key Factors That Affect Matrix of Linear Transformation Results

• The Rule of the Transformation: The most crucial factor is how the linear transformation T itself is defined. Different rules (like rotation, scaling, projection, shear) result in different images T(e_i) and thus different matrices.
• Domain Dimension (n): This determines the number of standard basis vectors (e₁, …, e_n) and thus the number of columns in the matrix A.
• Codomain Dimension (m): This determines the number of components in each image vector T(e_i) and thus the number of rows in the matrix A.
• Choice of Bases: While this calculator uses standard bases, the matrix representation of a linear transformation depends on the bases chosen for both the domain and codomain. A different choice of bases would yield a different matrix for the same transformation.
• The Specific Components of T(e_i): The numerical values you enter for the components of the images of the basis vectors directly form the columns of the matrix. Small changes here directly alter the matrix.
• Linearity of T: The entire process relies on T being a linear transformation. If T were not linear, it wouldn’t be representable by a single matrix in this way, and the concept of finding “the” matrix wouldn’t apply directly.

Using the matrix of linear transformation calculator accurately requires correctly identifying the images of the standard basis vectors under T.

Frequently Asked Questions (FAQ)

What is a linear transformation?

A linear transformation is a function T: V → W between vector spaces V and W that satisfies T(u+v) = T(u)+T(v) and T(cu) = cT(u) for all vectors u, v in V and scalar c.

Why is the matrix of a linear transformation important?

It allows us to represent the action of the transformation as matrix multiplication (T(v) = Av), which is computationally convenient and allows us to use tools from matrix theory to analyze the transformation (e.g., find kernel, range, eigenvalues).

What are standard basis vectors?

In Rⁿ, the standard basis vectors e₁, e₂, …, e_n are vectors with a 1 in the i-th position and 0s elsewhere. For R², they are [1,0]^T and [0,1]^T. For R³, they are [1,0,0]^T, [0,1,0]^T, and [0,0,1]^T.

Can every linear transformation be represented by a matrix?

Every linear transformation between finite-dimensional vector spaces can be represented by a matrix with respect to chosen bases for the domain and codomain. Our matrix of linear transformation calculator uses standard bases.

What if the transformation is not from Rⁿ to R^m?

If the vector spaces are finite-dimensional but not Rⁿ and R^m (e.g., spaces of polynomials), we can still find a matrix representation once we choose bases for both spaces and find the coordinates of the images of the domain basis vectors relative to the codomain basis.

How does changing the basis affect the matrix?

If you change the bases for the domain or codomain, the matrix representing the same linear transformation will generally change through a change-of-basis matrix multiplication.

What does it mean if the matrix is invertible?

If the matrix of a linear transformation T: Rⁿ → Rⁿ is invertible, it means the transformation is one-to-one and onto (an isomorphism), and there exists an inverse transformation T^-1 represented by the inverse matrix A^-1.

Can I use this matrix of linear transformation calculator for transformations involving complex numbers?

This calculator is designed for real vector spaces (Rⁿ, R^m). Linear transformations between complex vector spaces (like Cⁿ) also have matrix representations, but the components would be complex numbers.

Related Tools and Internal Resources

• Matrix Multiplication Calculator: Useful for applying the transformation matrix to a vector.
• Determinant Calculator: Find the determinant of the transformation matrix (if square) to understand scaling effects.
• Inverse Matrix Calculator: Find the inverse of the transformation matrix if it exists.
• Eigenvalue and Eigenvector Calculator: Analyze the directions unchanged (up to scaling) by the transformation.
• Vector Addition/Subtraction Calculator: Basic vector operations.
• Dot Product Calculator: Calculate the dot product of vectors.

These tools can help you further explore the properties and applications of the matrix obtained from our matrix of linear transformation calculator.