Unit 9 · Topic 4.12 · Optional
Linear Transformations and Matrices
Interpret a \(2\times2\) matrix as a function that maps every input vector in the plane to an output vector while preserving vector addition and scalar multiplication.
Learning Goals
- Evaluate a linear transformation using \(L(\mathbf v)=A\mathbf v\).
- Interpret each output component as a linear combination of input components.
- Use the origin and linearity properties to distinguish linear transformations.
- Calculate outputs for dilations, reflections, rotations, and shears.
- Transform several vectors at once by arranging them as matrix columns.
- Connect determinant information with area scaling and dimension collapse.
1. A Transformation Is a Function on Vectors
A transformation \(L:\mathbb R^2\to\mathbb R^2\) receives an input vector and returns an output vector. A \(2\times2\) matrix A defines
The input and output both have two components, but the matrix may change their magnitude, direction, or both.
2. Read the Matrix as Two Output Rules
For
matrix multiplication gives
Row 1 controls the new horizontal component; row 2 controls the new vertical component.
3. The Two Linearity Properties
Together these imply \(L(r\mathbf u+s\mathbf v)=rL(\mathbf u)+sL(\mathbf v)\). Matrix multiplication automatically satisfies both properties.
4. The Origin Must Stay Fixed
This is a necessary test for linearity. A rule such as \(T(x,y)=(x+3,y-1)\) moves the origin to \((3,-1)\), so it is a translation but not a linear transformation.
5. Input and Output Vectors
A transformation changes the vector from its preimage to its image. The arrow's tail remains at the origin for a linear transformation, while the matrix determines the new endpoint.
6. Worked Output Example
Let
Then
The image of \(\langle4,2\rangle\) is \(\langle6,10\rangle\).
7. Track Dimensions and Order
| Factor | Dimensions | Role |
|---|---|---|
| A | \(2\times2\) | Transformation |
| \(\mathbf v\) | \(2\times1\) | Input vector |
| \(A\mathbf v\) | \(2\times1\) | Output vector |
The product \(\mathbf vA\) is not defined when v is written as a column. The transformation matrix acts from the left.
8. Dilations and Reflections
| Matrix | Effect on \((x,y)\) |
|---|---|
| \(\begin{bmatrix}a&0\\0&b\end{bmatrix}\) | Horizontal scale a, vertical scale b |
| \(\begin{bmatrix}1&0\\0&-1\end{bmatrix}\) | Reflect across the x-axis |
| \(\begin{bmatrix}-1&0\\0&1\end{bmatrix}\) | Reflect across the y-axis |
| \(\begin{bmatrix}0&1\\1&0\end{bmatrix}\) | Reflect across \(y=x\) |
For example, \(\begin{bmatrix}2&0\\0&\frac12\end{bmatrix}\begin{bmatrix}3\\4\end{bmatrix}=\begin{bmatrix}6\\2\end{bmatrix}\).
9. A \(90^\circ\) Counterclockwise Rotation
Thus, \((2,1)\mapsto(-1,2)\). The input and output both have magnitude \(\sqrt5\), and their dot product \(2(-1)+1(2)=0\) confirms a right-angle turn.
10. Shears
A shear slides points by an amount proportional to the other coordinate:
With \(k=2\), \((1,3)\mapsto(7,3)\). The y-coordinate stays fixed, while x shifts farther for points with larger y-values.
11. Transform an Entire Shape
Represent each vertex as a vector, apply the same matrix, and reconnect the image vertices in their original order. Linear transformations keep line segments straight and preserve parallel lines, although lengths and angles need not remain unchanged.
12. Transform Many Vectors at Once
Place n input vectors as columns of a \(2\times n\) matrix. One multiplication transforms every column:
The result remains \(2\times n\), so each output column corresponds to the input column in the same position.
13. Batch Example: Shear a Rectangle
Use the rectangle vertices \((0,0),(2,0),(2,1),(0,1)\) and a horizontal shear:
The image vertices \((0,0),(2,0),(4,1),(2,1)\) form a parallelogram. The column order keeps the edges connected correctly.
14. Determinant Predicts Area Behavior
| Determinant information | Effect |
|---|---|
| \(|\det A|=1\) | Area is preserved |
| \(|\det A|>1\) | Area is enlarged by that factor |
| \(0<|\det A|<1\) | Area is compressed |
| \(\det A=0\) | Plane shapes collapse to zero area |
The shear matrix in the previous example has determinant 1, so the rectangle and image parallelogram have equal area. A zero determinant means distinct inputs may share an output and the transformation cannot be reversed.
15. AP Workflow and Common Errors
- Write each vector as a \(2\times1\) column.
- Confirm that the \(2\times2\) matrix multiplies from the left.
- Compute each output component as a row-column dot product.
- Interpret the new endpoint geometrically.
- For several vectors, preserve their column order.
- Check that a claimed linear rule maps the origin to itself.
- Use determinant information to interpret area and reversibility.
- Do not multiply corresponding entries only.
- Do not reverse the order to \(\mathbf vA\).
- Do not add a translation vector and still call the rule linear.
- Do not assume every linear transformation preserves length or angle.
- Do not treat rows as separate input vectors in a \(2\times n\) batch.
- Do not reconnect transformed vertices in a different order.
Key Takeaways
- Represent rotations, reflections, dilations, and shears as matrix transformations.
- Core relationship: \(R_{90}=\begin{pmatrix}0&-1\\1&0\end{pmatrix}\)
- Error check: Keep the sign pattern of the rotation matrix in the correct order.
- For \(A=\begin{bmatrix}3&1\\-2&4\end{bmatrix}\), find the image of \(\langle2,-1\rangle\).
- Explain why \(T(x,y)=(2x-y,x+3y)\) is linear but \(S(x,y)=(2x-y+1,x+3y)\) is not.
- Use the appropriate reflection matrix to map \((-3,5)\) across the y-axis.
- Apply a \(90^\circ\) counterclockwise rotation to all columns of \(\begin{bmatrix}1&0&-2\\2&3&1\end{bmatrix}\).
- Predict how the matrix \(\begin{bmatrix}2&0\\0&3\end{bmatrix}\) changes the area of a figure, then verify using its determinant.