Linear Transformation Calculator

Math Linear Algebra • Linear Transformations and Eigenvalues

Written by STEM Calculators Team

Apply a linear transformation \(T\) (represented by a matrix \(A\)) to a vector \(v\), computing \(Tv=Av\). Includes a 2D/3D before/after visualization, an animation \(A(t)=(1-t)I+tA\), and a linearity check.

Dimension

Tolerance

Display decimals

Steps

Show transformed grid / basis Show det / area-volume scale Show linearity check Show animation

Transformation matrix \(A\)

A is 2×2

Matrix entries accept decimals, scientific notation, fractions (e.g. 7/3), pi, e.

Vector \(v\)

Compute \(Tv = Av\).

Animation \(A(t)=(1-t)I+tA\)

At \(t=0\): identity. At \(t=1\): full transformation.

Progress

Results

Output \(Tv\)

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Check \(Av\) (components)

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\(\det(A)\) / scale

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Notes

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Ready

Before / after visualization

Shows \(v\) and \(Tv\). If enabled, also shows the image of the unit basis and a small grid. Axis ticks are numbered and scaling is equal.

2D: drag to pan • wheel to zoom • double-click to reset view

Enter \(A\) and \(v\), then click “Calculate”.

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Theory

Linear transformation and matrix form

A function \(T:\mathbb{R}^n\to\mathbb{R}^n\) is a linear transformation if it satisfies: \[ T(u+v)=T(u)+T(v),\qquad T(\alpha v)=\alpha T(v) \] for all vectors \(u,v\) and scalars \(\alpha\).

Every linear transformation on \(\mathbb{R}^n\) can be represented by a matrix \(A\) so that \[ T(v)=Av. \] This calculator applies that rule directly (matrix–vector multiplication).

How to compute \(Tv=Av\)

If \(A=(a_{ij})\) and \(v=(v_1,\dots,v_n)^T\), then the output vector \(Tv\) has components: \[ (Tv)_i = \sum_{j=1}^n a_{ij}v_j. \] In words: each output entry is the dot product of a row of \(A\) with the vector \(v\).

Geometric meaning: basis vectors and grids

The columns of \(A\) tell you where the standard basis vectors go: \[ A e_1 = \text{col}_1(A),\quad A e_2 = \text{col}_2(A),\ \dots \] Because a linear map is determined by its action on a basis, visualizing \(Ae_i\) explains the whole transformation.

In 2D, transforming the unit square shows how area and orientation change. In 3D, transforming the unit cube shows how volume changes.

Determinant: area/volume scale and orientation

For \(A\in\mathbb{R}^{2\times 2}\), \(\det(A)\) is the signed area scale factor. For \(A\in\mathbb{R}^{3\times 3}\), \(\det(A)\) is the signed volume scale factor.

If \(\det(A)=0\), the transformation is singular (it flattens space to lower dimension).
If \(\det(A)<0\), orientation flips (a reflection is involved).
If \(|\det(A)|>1\), areas/volumes expand; if \(|\det(A)|<1\), they shrink.

Linearity check

If \(T(v)=Av\), then \(T\) is automatically linear, and it always satisfies: \[ T(a v + b w)=aT(v)+bT(w). \] The calculator includes a numerical check (useful to catch input mistakes or confirm understanding).

Animation idea: \(A(t)=(1-t)I+tA\)

To animate smoothly from “no transformation” to the full transformation, we use \[ A(t)=(1-t)I+tA,\qquad 0\le t\le 1. \] At \(t=0\), the map is the identity. At \(t=1\), it equals \(A\). This gives an intuitive deformation of vectors and grids.

University note: operator norms and eigenvalues

In advanced courses, you may measure how much a linear operator can stretch vectors using an operator norm, e.g. \[ \|A\|=\max_{\|v\|=1}\|Av\|. \] Eigenvalues and eigenvectors describe special directions where the transformation only scales: \(Av=\lambda v\). This chapter often continues from applying \(A\) to vectors to analyzing eigen-structure.

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