Find The Standard Matrix Of The Linear Transformation Calculator

Easily find the standard matrix for a linear transformation T: Rⁿ → R^m by providing the images of the standard basis vectors. Our calculator simplifies the process and gives you the matrix directly.

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Standard Matrix A:

Images of Standard Basis Vectors:

Formula Used:

The standard matrix A of a linear transformation T: Rⁿ → R^m is the m x n matrix whose columns are the images of the standard basis vectors of Rⁿ under T: A = [T(e₁) T(e₂) … T(e_n)], where e_i are the standard basis vectors of Rⁿ.

What is the Standard Matrix of a Linear Transformation?

The standard matrix of a linear transformation is a unique matrix that represents a linear transformation T from Rⁿ to R^m with respect to the standard bases of Rⁿ and R^m. If A is the standard matrix of T, then applying the transformation T to a vector x in Rⁿ is equivalent to multiplying x by the matrix A: T(x) = Ax.

This matrix is incredibly useful because it allows us to represent a linear transformation (which is a function between vector spaces) as a matrix, and the action of the transformation as matrix multiplication. The columns of the standard matrix of a linear transformation are simply the images of the standard basis vectors of the domain Rⁿ under the transformation T.

Anyone studying linear algebra, computer graphics, physics, engineering, or any field involving vector transformations will find the concept of the standard matrix of a linear transformation essential. It simplifies the process of applying transformations and understanding their geometric effects.

A common misconception is that every function between vector spaces has a standard matrix. Only *linear* transformations have standard matrices. A transformation T is linear if T(cu + dv) = cT(u) + dT(v) for all scalars c, d and vectors u, v.

Standard Matrix of a Linear Transformation Formula and Mathematical Explanation

Let T: Rⁿ → R^m be a linear transformation. The standard basis for Rⁿ consists of vectors e₁, e₂, …, e_n, where e_i is a vector with a 1 in the i-th position and 0s elsewhere.

For example, in R², e₁ = (1, 0) and e₂ = (0, 1). In R³, e₁ = (1, 0, 0), e₂ = (0, 1, 0), and e₃ = (0, 0, 1).

To find the standard matrix of a linear transformation T, we apply T to each of these standard basis vectors:

• T(e₁) = v₁
• T(e₂) = v₂
• …
• T(e_n) = v_n

Each T(e_i) will be a vector in R^m. The standard matrix A of T is the m x n matrix whose columns are these vectors v₁, v₂, …, v_n:

A = [ v₁ | v₂ | … | v_n ] = [ T(e₁) | T(e₂) | … | T(e_n) ]

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

A =
[ a₁₁ a₁₂ … a_1n ]
[ a₂₁ a₂₂ … a_2n ]
[ … … … … ]
[ a_m1 a_m2 … a_mn ]

Variable	Meaning	Type	Typical Range
n	Dimension of the domain R^n	Positive Integer	1, 2, 3, 4…
m	Dimension of the codomain R^m	Positive Integer	1, 2, 3, 4…
ei	i-th standard basis vector of R^n	Vector	Vector with 1 in i-th place, 0 elsewhere
T(ei)	Image of ei under T, a vector in R^m	Vector	Components are real numbers
A	The m x n standard matrix of T	Matrix	Entries are real numbers

Table explaining the variables involved in finding the standard matrix.

Practical Examples (Real-World Use Cases)

Example 1: Rotation in R²

Let T: R² → R² be a linear transformation that rotates vectors counterclockwise by 90 degrees. The standard basis vectors in R² are e₁ = (1, 0) and e₂ = (0, 1). T(e₁) = T(1, 0) = (0, 1) T(e₂) = T(0, 1) = (-1, 0) The standard matrix of this linear transformation is A = [ T(e₁) | T(e₂) ] = [ 0 -1 ]
[ 1 0 ]

Example 2: Projection onto the xy-plane in R³

Let T: R³ → R³ be a projection onto the xy-plane. So, T(x, y, z) = (x, y, 0). The standard basis vectors in R³ are e₁ = (1, 0, 0), e₂ = (0, 1, 0), and e₃ = (0, 0, 1). T(e₁) = T(1, 0, 0) = (1, 0, 0) T(e₂) = T(0, 1, 0) = (0, 1, 0) T(e₃) = T(0, 0, 1) = (0, 0, 0) The standard matrix of this linear transformation is A = [ T(e₁) | T(e₂) | T(e₃) ] = [ 1 0 0 ]
[ 0 1 0 ]
[ 0 0 0 ]

How to Use This Standard Matrix of Linear Transformation Calculator

1. Select Dimensions: Choose the dimension of the domain (n) and the dimension of the codomain (m) from the dropdown menus. The calculator supports dimensions from 1 to 4.
2. Enter Image Vector Components: For each standard basis vector e_i of the domain Rⁿ (where i goes from 1 to n), input the components of its image T(e_i) in R^m. Input fields will appear dynamically based on your chosen n and m.
3. Calculate: Click the "Calculate Matrix" button.
4. View Results: The calculator will display the standard matrix of the linear transformation A, along with the image vectors T(e_i) you entered.
5. Interpret Matrix: The resulting m x n matrix A is the standard matrix. Multiplying this matrix by any vector x in Rⁿ will give you T(x).
6. Chart (if n ≥ 2): If the domain dimension n is 2 or more, a chart will visualize the components of T(e₁) and T(e₂).
7. Reset: Use the "Reset" button to clear inputs and return to default dimensions (n=2, m=2).
8. Copy: Use the "Copy Results" button to copy the matrix and input vectors.

Understanding the resulting standard matrix of a linear transformation allows you to quickly apply the transformation to any vector and analyze its properties.

Key Factors That Affect the Standard Matrix of a Linear Transformation Results

• Dimension of Domain (n): The number of columns in the standard matrix is equal to n. Changing n changes the number of standard basis vectors whose images are needed.
• Dimension of Codomain (m): The number of rows in the standard matrix is equal to m. Changing m changes the number of components in each image vector T(e_i).
• Images of Standard Basis Vectors (T(e_i)): These vectors form the columns of the standard matrix. Any change in how the transformation T acts on the basis vectors directly changes the matrix.
• The Transformation Rule T Itself: The fundamental definition of the linear transformation T dictates what T(e_i) will be. Different transformations (rotation, projection, scaling, shear) will have vastly different standard matrices.
• Choice of Basis (Though Standard Here): While this calculator finds the matrix with respect to the *standard* bases, using different bases for Rⁿ and R^m would result in a different matrix for the same linear transformation.
• Linearity of the Transformation: The very concept of a standard matrix only applies to *linear* transformations. If the transformation is not linear, it cannot be represented by a single matrix multiplication in this way.

Frequently Asked Questions (FAQ)

What is a linear transformation?

A transformation (or function) T between vector spaces is linear if it preserves vector addition and scalar multiplication: T(u + v) = T(u) + T(v) and T(cu) = cT(u) for all vectors u, v and scalar c.

Why is it called the 'standard' matrix?

It's called the standard matrix because it's derived using the standard basis vectors (e₁, e₂, …) in both the domain and codomain (implicitly for the codomain when we write T(e_i) as column vectors).

How do I know the images T(e_i) for my transformation?

You need to know how the transformation acts. For example, if T is a rotation by θ in R², T(1,0)=(cos θ, sin θ) and T(0,1)=(-sin θ, cos θ). If T is defined by a formula T(x,y) = (x+y, x-y), then T(1,0)=(1,1) and T(0,1)=(1,-1).

Can every matrix be a standard matrix of some linear transformation?

Yes, any m x n matrix A can be considered the standard matrix of the linear transformation T: Rⁿ → R^m defined by T(x) = Ax.

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

The concept of a standard matrix as defined here is specifically for linear transformations between Rⁿ and R^m using standard bases. More general linear transformations between other vector spaces have matrix representations, but they depend on the chosen bases for those spaces.

What does the determinant of the standard matrix tell me (if n=m)?

If n=m, the standard matrix is square. Its determinant tells you about how the transformation scales areas (in R²) or volumes (in R³), and whether it preserves orientation (sign of the determinant).

What if T(e_i) are linearly dependent?

If the columns of the standard matrix (the vectors T(e_i)) are linearly dependent, the transformation T is not one-to-one, and its range is a subspace of R^m with dimension less than n (if n ≤ m).

Can I use this calculator for transformations involving polynomials or other vector spaces?

No, this calculator is specifically for linear transformations T: Rⁿ → R^m, where vectors are columns of real numbers and we use the standard bases. Finding matrix representations for transformations on other vector spaces like polynomial spaces requires choosing bases for those spaces first.

Related Tools and Internal Resources

• Matrix Multiplication Calculator: Useful for applying the calculated standard matrix to vectors.
• Determinant Calculator: Find the determinant of the standard matrix if it's square.
• Eigenvalue and Eigenvector Calculator: Analyze the properties of the linear transformation through its eigenvalues and eigenvectors (if n=m).
• Vector Addition/Subtraction Calculator: Perform basic vector operations.
• Dot Product Calculator: Calculate the dot product of vectors.
• Cross Product Calculator: Calculate the cross product of 3D vectors.