Linear Transformation

Linear Transformation

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Prerequisites

Basis Vector
Span
Linear independent/dependent

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What is Linear Transformation

1. What is Transformatin?

Any linear transformation in 2D can be written as:

\[A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\]

The columns of the matrix tell us:

• Left column → where $\hat{i}$ lands
• Right column → where $\hat{j}$ lands

Because:

\[A\hat{i} = A \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} a \\ c \end{bmatrix}\] \[A\hat{j} = A \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} b \\ d \end{bmatrix}\]

So:

\[A = \begin{bmatrix} \; A\hat{i} \;|\; A\hat{j} \; \end{bmatrix}\]

Matrix = transformed basis vectors.

This A matrix transform Identiy space(usual 2D coordination systems) to A space. Take any vector:

\[\vec{x} = \begin{bmatrix} x \\ y \end{bmatrix}\]

We can rewrite it as:

\[\vec{x} = x\hat{i} + y\hat{j}\]

Now apply transformation:

\[A\vec{x} = A(x\hat{i} + y\hat{j})\]

2. What if the matrix is linear dependent

Suppose:

\[\text{column}_2 = c \cdot \text{column}_1\]

Then:

\[A = \begin{bmatrix} \vec{v} & c\vec{v} \end{bmatrix}\]

Now compute:

\[A \begin{bmatrix} x \\ y \end{bmatrix} = x\vec{v} + y(c\vec{v}) = (x + cy)\vec{v}\]

Everything lies along one direction. Instead of mapping to a 2D plane, all outputs lie on a single line The transformation is called Compresses dimension(2D → 1D) or Rank deficiency

Additional: Determinant Connection

If columns are dependent:

\[\det(A) = 0\]

Because area collapses to zero. we already see the Compresses dimesion 2D → 1D, a square becomes a line. It means area is to become 0. So determinant measures: Area scaling factor.

3. Matrix Muliplication as Composition of Transformations

A matrix represents a linear transformation.

If we have two matrices:

• $ M_1 $
• $ M_2 $

Applying them sequentially to a vector ( x ):

\[x \xrightarrow{M_1} M_1x \xrightarrow{M_2} M_2(M_1x)\]

This entire process can be written as:

\[M_2(M_1x) = (M_2M_1)x\]

Matrix multiplication represents composition of transformations.

Insight of Composition

Let

\[M_2 = \begin{bmatrix} a & b \\ c & d \end{bmatrix} , \quad M_1 = \begin{bmatrix} e & f \\ g & h \end{bmatrix}\]

The columns of $ M_1 $ are:

\[\text{col}_1 = \begin{bmatrix} e \\ g \end{bmatrix} , \quad \text{col}_2 = \begin{bmatrix} f \\ h \end{bmatrix}\]

The product $ M_2M_1 $ is formed by transforming each column of $ M_1 $ by $ M_2 $:

\[M_2M_1 = \begin{bmatrix} M_2\text{col}_1 & M_2\text{col}_2 \end{bmatrix}\]

In other words:

• First column of result = $ M_2 $ applied to first column of $ M_1 $
• Second column of result = $ M_2 $ applied to second column of $ M_1 $

It is also related to change of basis from previous post.

With Example: Rotation + Shear as Composition

Rotation by 90° (counterclockwise)

\[R = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\]

Shear in x-direction (parameter $ k $)

\[S = \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}\]

We want: First Rotate, Then Shear

So transformation is:

\[T = SR\]

Because:

\[Tx = S(Rx)\]

Compute $ SR $

\[SR = \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} k & -1 \\ 1 & 0 \end{bmatrix}\]

So the combined transformation is:

\[T = \begin{bmatrix} k & -1 \\ 1 & 0 \end{bmatrix}\]

It means a matrix is not just one transformation, sometimes composed of over 2 matrix transformations. So later we can see decomposition method. It can connect with the perspective easily.

4. Properties of composition

1. Non-Commutativity

\[M_1M_2 \ne M_2M_1\]

Because Rotate → Shear is NOT the same as Shear → Rotate

The order changes the geometric result. Matrix multiplication is not commutative.

1. Associativity

\[(AB)C = A(BC)\]

This is always true. Applying three transformations in sequence:

\[x \xrightarrow{C} Cx \xrightarrow{B} B(Cx) \xrightarrow{A} A(B(Cx))\]

No matter how we group them, the final transformation is:

\[A(B(Cx))\]

• Commutative ❌
• Associative ✅