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AlgebraFree

Linear Transformation Calculator

Use this free linear transformation calculator to apply a 2x2 matrix to a vector or analyze transformation properties. Get the transformed vector, determinant (area scaling), trace, eigenvalues, and whether the transformation preserves or reverses orientation.

Enter Values

Apply transforms a specific vector. Properties analyzes the matrix itself.

Top-left entry of the transformation matrix

Top-right entry of the transformation matrix

Bottom-left entry of the transformation matrix

Bottom-right entry of the transformation matrix

The x component of the input vector (used in apply mode)

The y component of the input vector (used in apply mode)

Result

Enter values above and click Calculate to see your result.

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Formula

T(v) = Av (matrix times vector)

In apply mode, multiply the 2x2 matrix A by the input vector v to get the transformed vector. In properties mode, compute determinant (area scaling), trace, eigenvalues, and classify the transformation type (rotation, scaling, shear, or reflection).

Worked Example

A = [[2, 0], [0, 3]], v = [1, 1], Mode: Apply Step 1: Result x = (2)(1) + (0)(1) = 2 Step 2: Result y = (0)(1) + (3)(1) = 3 Result: T(v) = [2, 3]. The matrix stretched x by 2 and y by 3.

What Is a Linear Transformation?

A linear transformation is a function between vector spaces that preserves addition and scalar multiplication. Every linear transformation from 2D to 2D can be represented by a 2x2 matrix. When you multiply a matrix by a vector, you are applying a linear transformation that can rotate, scale, shear, reflect, or project the vector.
  • The columns of the matrix tell you where the standard basis vectors (i and j) end up after the transformation
  • The determinant measures how the transformation scales area: det = 2 means areas double, det = 0 means the transformation collapses to a line
  • A negative determinant means the transformation includes a reflection (orientation reversal)
  • Common transformations include rotation (cos/sin matrix), scaling (diagonal matrix), shear (off-diagonal entries), and projection
  • Composing two transformations corresponds to multiplying their matrices

Linear transformations are the foundation of computer graphics, robotics, signal processing, and quantum mechanics. Understanding how matrices transform vectors is essential for working with coordinate systems, animations, and physical simulations.

You can also calculate changes using our Eigenvalue Calculator, 2x2 Determinant Calculator, Vector Projection Calculator or Matrix Multiplication Calculator.

Frequently Asked Questions

What does the determinant mean for a transformation?

The absolute value of the determinant is the area scaling factor. If det = 3, every shape's area triples. If det = 0.5, areas halve. If det = 0, the transformation collapses 2D space to a line or point. A negative sign means orientation is reversed (like a mirror).

How do I create a rotation matrix?

A rotation by angle theta uses the matrix [[cos(theta), -sin(theta)], [sin(theta), cos(theta)]]. For 90 degrees: [[0, -1], [1, 0]]. For 45 degrees: [[0.707, -0.707], [0.707, 0.707]].

What is a shear transformation?

A shear slides points parallel to one axis. A horizontal shear is [[1, k], [0, 1]] where k controls the shear amount. A vertical shear is [[1, 0], [k, 1]]. Shear preserves area (det = 1) but changes angles.

Why does properties mode ignore the vector inputs?

Transformation properties (determinant, trace, eigenvalues, type) depend only on the matrix itself, not on any particular input vector. They describe how the transformation behaves on all vectors simultaneously.

How do I compose multiple transformations?

Multiply the matrices in reverse order. If you want to first apply A, then B, the combined matrix is BA (B times A). Matrix multiplication is not commutative, so the order matters.

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Eigenvalue Calculator2x2 Determinant CalculatorVector Projection CalculatorMatrix Multiplication Calculator

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