What is a Linear Transformation?

A linear transformation is a function between two vector spaces that preserves vector addition and scalar multiplication. In simpler terms, it’s a way of transforming vectors that maintains certain mathematical properties. Linear transformations are fundamental in linear algebra and have numerous applications in physics, computer graphics, and data analysis.

Mathematically, a transformation T from vector space V to vector space W is linear if for any vectors u and v in V and any scalar c:

T(u + v) = T(u) + T(v)

T(cu) = cT(u)

Linear transformations can be represented by matrices, which is why they’re often called matrix transformations.

How to Use the Linear Transformation Calculator

Step 1: Enter the Transformation Matrix

In the first input field, enter your transformation matrix. Use commas to separate elements within a row and semicolons to separate rows. For example, a 2×2 matrix would be entered as “1,2;3,4”.

Step 2: Input the Vector

In the second input field, enter the vector you want to transform. Use commas to separate the vector components. For instance, a 2D vector would be entered as “1,2”.

Step 3: Click “Calculate Transformation”

After entering both the matrix and vector, click the “Calculate Transformation” button. The calculator will process your input and display the result.

Step 4: Interpret the Results

The calculator will show the transformed vector as the result. It will also provide a brief explanation of how the transformation was calculated, helping you understand the process.

Step 5: Analyze the Transformation

Consider how the original vector has been changed by the transformation. Has it been rotated, scaled, or both? This can give you insights into the nature of the linear transformation represented by your matrix.