Interactive Linear Algebra: Matrix-Vector Multiplication
Visualize 2D matrix-vector multiplication with interactive animations. Understand linear transformations through drag-and-drop vector manipulation and real-time matrix operations.
Understanding Matrix-Vector Multiplication
Matrix-vector multiplication is one of the most fundamental operations in linear algebra. In this interactive tutorial, you'll see exactly how a 2×2 matrix transforms a 2D vector through visual animations.
Key Concept
When we multiply a matrix by a vector, we're applying a linear transformation. The matrix acts as a function that takes one vector and produces another vector, potentially changing both its direction and magnitude.
Interactive 2D Matrix-Vector Multiplication
Drag the vector endpoint (blue dot) to see how the matrix transforms it in real-time. The red vector shows the result of the matrix multiplication.
Matrix A (2×2)
Vector v
Mathematical Foundation
The multiplication of a 2×2 matrix A by a 2D vector v is defined as:
a₂₁ a₂₂
vᵧ
a₂₁vₓ + a₂₂vᵧ
Step-by-Step Calculation
Step 1: Identify Matrix and Vector Elements
From our current matrix and vector:
Step 2: Apply the Formula
Calculate each component of the result vector:
Step 3: Result
The transformed vector is:
Visual Interpretation
Rotation
Matrices like [[0, -1], [1, 0]] rotate vectors counterclockwise by 90°.
Scaling
Matrices like [[2, 0], [0, 2]] scale vectors by a factor of 2 in both directions.
Shearing
Matrices like [[1, 1], [0, 1]] shear vectors, tilting them horizontally.
Reflection
Matrices like [[-1, 0], [0, 1]] reflect vectors across the y-axis.
Practice Problems
Problem 1: Rotation Matrix
Set the matrix to rotate by 45° and observe how it transforms the unit vector [1, 0].
Problem 2: Scaling Matrix
Create a matrix that scales by 3 in the x-direction and 0.5 in the y-direction.
Problem 3: Combined Transformation
Experiment with matrices that combine rotation and scaling.
Key Takeaways
- Linear Transformation: Matrix-vector multiplication represents a linear transformation of space.
- Geometric Meaning: Each matrix corresponds to a specific geometric transformation (rotation, scaling, shearing, reflection).
- Determinant: The determinant tells us about the area scaling factor and whether the transformation preserves orientation.
- Composition: Multiple transformations can be combined by multiplying their matrices.
- Reversibility: If the determinant is non-zero, the transformation can be reversed using the inverse matrix.