The Central Idea
A matrix can transform a vector by multiplication.
If
\[ A= \begin{bmatrix} a & b\\ c & d \end{bmatrix} \]and
\[ \mathbf{v} = \begin{bmatrix} x\\ y \end{bmatrix}, \]then
\[ \boxed{ A\mathbf{v} = \begin{bmatrix} ax+by\\ cx+dy \end{bmatrix} }. \]Different matrices produce different geometric effects.
Matrices as Rules for Space
A transformation matrix does not merely store numbers.
It defines a rule that sends each input vector to an output vector.
Symbolically,
\[ \boxed{ \mathbf{v} \longmapsto A\mathbf{v} }. \]Applying the matrix to every point or position vector of a figure transforms the entire figure.
The Identity Matrix
The simplest transformation matrix is
\[ I= \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix}. \]For every vector
\[ \mathbf{v} = \begin{bmatrix} x\\ y \end{bmatrix}, \]we have
\[ \boxed{ I\mathbf{v} = \mathbf{v} }. \]The identity transformation changes nothing.
Why the Columns Matter
The columns of a transformation matrix show where the standard basis vectors go.
Recall
\[ \mathbf{i} = \begin{bmatrix} 1\\ 0 \end{bmatrix}, \qquad \mathbf{j} = \begin{bmatrix} 0\\ 1 \end{bmatrix}. \]If
\[ A= \begin{bmatrix} a&b\\ c&d \end{bmatrix}, \]then
\[ A\mathbf{i} = \begin{bmatrix} a\\ c \end{bmatrix}, \qquad A\mathbf{j} = \begin{bmatrix} b\\ d \end{bmatrix}. \]The Matrix Is Determined by the Basis Directions
Every plane vector can be written
\[ \mathbf{v} = x\mathbf{i}+y\mathbf{j}. \]A matrix transformation acts linearly:
\[ A\mathbf{v} = xA\mathbf{i}+yA\mathbf{j}. \]Therefore, once we know where \(\mathbf{i}\) and \(\mathbf{j}\) go, we know how the matrix acts on every vector in the plane.
Rotation about the Origin
A counterclockwise rotation through angle \(\theta\) about the origin is represented by
\[ \boxed{ R_\theta = \begin{bmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta \end{bmatrix} }. \]This transformation changes direction while preserving distance from the origin.
Why the Rotation Matrix Has This Form
The standard unit vector
\[ \mathbf{i} = \begin{bmatrix} 1\\ 0 \end{bmatrix} \]rotates to
\[ \begin{bmatrix} \cos\theta\\ \sin\theta \end{bmatrix}. \]Likewise,
\[ \mathbf{j} = \begin{bmatrix} 0\\ 1 \end{bmatrix} \]rotates to
\[ \begin{bmatrix} -\sin\theta\\ \cos\theta \end{bmatrix}. \]These two images become the columns of \(R_\theta\).
Rotation of a General Vector
Applying
\[ R_\theta = \begin{bmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta \end{bmatrix} \]to
\[ \mathbf{v} = \begin{bmatrix} x\\ y \end{bmatrix} \]gives
\[ \boxed{ R_\theta\mathbf{v} = \begin{bmatrix} x\cos\theta-y\sin\theta\\ x\sin\theta+y\cos\theta \end{bmatrix} }. \]Rotation by \(90^\circ\)
For
\[ \theta=90^\circ, \]we have
\[ \cos90^\circ=0, \qquad \sin90^\circ=1. \]Therefore,
\[ \boxed{ R_{90^\circ} = \begin{bmatrix} 0&-1\\ 1&0 \end{bmatrix} }. \]It sends
\[ (x,y)\mapsto(-y,x). \]Example: Rotate by \(90^\circ\)
Rotate
\[ \mathbf{v} = \begin{bmatrix} 3\\ 2 \end{bmatrix} \]counterclockwise by \(90^\circ\).
\[ R_{90^\circ}\mathbf{v} = \begin{bmatrix} 0&-1\\ 1&0 \end{bmatrix} \begin{bmatrix} 3\\ 2 \end{bmatrix}. \]Thus,
\[ \boxed{ R_{90^\circ}\mathbf{v} = \begin{bmatrix} -2\\ 3 \end{bmatrix} }. \]Rotation by \(180^\circ\)
Since
\[ \cos180^\circ=-1, \qquad \sin180^\circ=0, \]we get
\[ \boxed{ R_{180^\circ} = \begin{bmatrix} -1&0\\ 0&-1 \end{bmatrix} }. \]Therefore,
\[ (x,y)\mapsto(-x,-y). \]This is also reflection through the origin.
Rotation by \(270^\circ\)
A counterclockwise rotation by \(270^\circ\) is equivalent to a clockwise rotation by \(90^\circ\).
\[ \boxed{ R_{270^\circ} = \begin{bmatrix} 0&1\\ -1&0 \end{bmatrix} }. \]It sends
\[ (x,y)\mapsto(y,-x). \]Rotation Preserves Magnitude
Rotating a vector changes its direction but not its length.
If
\[ \mathbf{w} = R_\theta\mathbf{v}, \]then
\[ \boxed{ \|\mathbf{w}\| = \|\mathbf{v}\| }. \]Rotations are therefore distance-preserving transformations.
Rotation Preserves Angles
If two vectors are rotated by the same rotation matrix, the angle between them remains unchanged.
A rigid rotation therefore preserves:
- lengths;
- angles;
- parallelism;
- perpendicularity;
- shape.
Reflection across the \(x\)-Axis
Reflection across the \(x\)-axis leaves the \(x\)-coordinate unchanged and reverses the sign of the \(y\)-coordinate.
\[ (x,y)\mapsto(x,-y). \]The matrix is
\[ \boxed{ F_x = \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix} }. \]Example: Reflect across the \(x\)-Axis
Let
\[ \mathbf{v} = \begin{bmatrix} -3\\ 5 \end{bmatrix}. \]Then
\[ F_x\mathbf{v} = \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix} \begin{bmatrix} -3\\ 5 \end{bmatrix}. \]Therefore,
\[ \boxed{ F_x\mathbf{v} = \begin{bmatrix} -3\\ -5 \end{bmatrix} }. \]Reflection across the \(y\)-Axis
Reflection across the \(y\)-axis sends
\[ (x,y)\mapsto(-x,y). \]Therefore,
\[ \boxed{ F_y = \begin{bmatrix} -1&0\\ 0&1 \end{bmatrix} }. \]Reflection across the Line \(y=x\)
Reflection across
\[ y=x \]exchanges the coordinates:
\[ (x,y)\mapsto(y,x). \]The matrix is
\[ \boxed{ F_{y=x} = \begin{bmatrix} 0&1\\ 1&0 \end{bmatrix} }. \]Reflection across the Line \(y=-x\)
Reflection across
\[ y=-x \]sends
\[ (x,y)\mapsto(-y,-x). \]Therefore,
\[ \boxed{ F_{y=-x} = \begin{bmatrix} 0&-1\\ -1&0 \end{bmatrix} }. \]Reflection through the Origin
Reflection through the origin sends
\[ (x,y)\mapsto(-x,-y). \]Its matrix is
\[ \boxed{ \begin{bmatrix} -1&0\\ 0&-1 \end{bmatrix} }. \]This is the same matrix as rotation by \(180^\circ\).
Reflections Preserve Length
Like rotations, ordinary reflections preserve distances.
If
\[ \mathbf{w} = F\mathbf{v}, \]where \(F\) is a reflection matrix, then
\[ \boxed{ \|\mathbf{w}\| = \|\mathbf{v}\| }. \]Reflections preserve shape but reverse orientation.
What Orientation Reversal Means
Imagine three noncollinear points listed in counterclockwise order.
After a reflection, their images appear in clockwise order.
The figure retains its lengths and angles, but its handedness is reversed.
Uniform Scaling
A uniform scaling multiplies every coordinate by the same scalar \(k\).
\[ (x,y)\mapsto(kx,ky). \]The matrix is
\[ \boxed{ S_k = \begin{bmatrix} k&0\\ 0&k \end{bmatrix} }. \]Example: Enlarge by a Factor of \(3\)
Let
\[ \mathbf{v} = \begin{bmatrix} 2\\ -1 \end{bmatrix}. \]Use
\[ S_3 = \begin{bmatrix} 3&0\\ 0&3 \end{bmatrix}. \]Then
\[ \boxed{ S_3\mathbf{v} = \begin{bmatrix} 6\\ -3 \end{bmatrix} }. \]Uniform Contraction
If
\[ 0For example,
\[ S_{1/2} = \begin{bmatrix} \frac12&0\\ 0&\frac12 \end{bmatrix} \]halves every distance from the origin.
Negative Uniform Scaling
If \(k<0\), uniform scaling changes magnitude by \(|k|\) and reverses every direction through the origin.
For example,
\[ S_{-2} = \begin{bmatrix} -2&0\\ 0&-2 \end{bmatrix}. \]This combines a \(180^\circ\) rotation with enlargement by a factor of \(2\).
Nonuniform Scaling
Horizontal and vertical directions may be scaled by different factors.
The general axis-aligned scaling matrix is
\[ \boxed{ S = \begin{bmatrix} a&0\\ 0&b \end{bmatrix} }. \]It sends
\[ (x,y)\mapsto(ax,by). \]Example: Different Horizontal and Vertical Scales
Let
\[ S= \begin{bmatrix} 2&0\\ 0&\frac12 \end{bmatrix}. \]Then
\[ (x,y)\mapsto \left( 2x,\frac12y \right). \]The transformation stretches horizontally and compresses vertically.
Nonuniform Scaling Can Change Angles
Uniform scaling preserves angles because all directions are scaled equally.
Nonuniform scaling generally does not preserve angles.
A square, for example, can become a rectangle when its horizontal and vertical directions are scaled differently.
Scaling and Area
If horizontal lengths are multiplied by \(a\) and vertical lengths by \(b\), areas are multiplied by
\[ \boxed{|ab|}. \]For a uniform scaling by \(k\), area scales by
\[ \boxed{k^2}. \]What Is a Shear?
A shear shifts one coordinate by an amount proportional to another coordinate.
Unlike a rotation or reflection, a shear usually changes angles and lengths.
Parallel lines remain parallel, but shapes become slanted.
Horizontal Shear
A horizontal shear with factor \(k\) sends
\[ (x,y)\mapsto(x+ky,y). \]Its matrix is
\[ \boxed{ H_k = \begin{bmatrix} 1&k\\ 0&1 \end{bmatrix} }. \]The \(y\)-coordinate controls how much horizontal shift occurs.
Example: Horizontal Shear
Let
\[ H_2 = \begin{bmatrix} 1&2\\ 0&1 \end{bmatrix} \]and
\[ \mathbf{v} = \begin{bmatrix} 3\\ 4 \end{bmatrix}. \]Then
\[ H_2\mathbf{v} = \begin{bmatrix} 3+2(4)\\ 4 \end{bmatrix}. \]Therefore,
\[ \boxed{ H_2\mathbf{v} = \begin{bmatrix} 11\\ 4 \end{bmatrix} }. \]Vertical Shear
A vertical shear with factor \(k\) sends
\[ (x,y)\mapsto(x,y+kx). \]Its matrix is
\[ \boxed{ V_k = \begin{bmatrix} 1&0\\ k&1 \end{bmatrix} }. \]The \(x\)-coordinate controls how much vertical shift occurs.
Example: Vertical Shear
Let
\[ V_{-1} = \begin{bmatrix} 1&0\\ -1&1 \end{bmatrix} \]and
\[ \mathbf{v} = \begin{bmatrix} 2\\ 5 \end{bmatrix}. \]Then
\[ V_{-1}\mathbf{v} = \begin{bmatrix} 2\\ -2+5 \end{bmatrix} = \boxed{ \begin{bmatrix} 2\\ 3 \end{bmatrix} }. \]What a Shear Does to the Basis
For the horizontal shear
\[ H_k = \begin{bmatrix} 1&k\\ 0&1 \end{bmatrix}, \]we have
\[ H_k\mathbf{i} = \mathbf{i} \]and
\[ H_k\mathbf{j} = \begin{bmatrix} k\\ 1 \end{bmatrix}. \]The horizontal basis direction stays fixed while the vertical basis direction tilts.
Shears Preserve Parallelism
A shear sends parallel lines to parallel lines.
This happens because linear combinations and scalar multiples are preserved by matrix transformations.
However, perpendicular lines need not remain perpendicular.
Shears Preserve Area in the Standard Case
The standard horizontal and vertical shear matrices
\[ \begin{bmatrix} 1&k\\ 0&1 \end{bmatrix} \]and
\[ \begin{bmatrix} 1&0\\ k&1 \end{bmatrix} \]preserve area.
A rectangle may become a parallelogram, but its area remains the same.
Transforming an Entire Figure
To transform a polygon or other coordinate figure, apply the same matrix to every vertex.
If a triangle has position vectors
\[ \mathbf{a},\qquad \mathbf{b},\qquad \mathbf{c}, \]then its transformed vertices are
\[ A\mathbf{a},\qquad A\mathbf{b},\qquad A\mathbf{c}. \]Example: Rotate a Triangle by \(90^\circ\)
Suppose the vertices are
\[ A=(1,0),\qquad B=(3,0),\qquad C=(1,2). \]Under
\[ R_{90^\circ} = \begin{bmatrix} 0&-1\\ 1&0 \end{bmatrix}, \]the coordinate rule is
\[ (x,y)\mapsto(-y,x). \]Complete the Triangle Rotation
Therefore,
\[ A=(1,0)\mapsto A'=(0,1), \] \[ B=(3,0)\mapsto B'=(0,3), \] \[ C=(1,2)\mapsto C'=(-2,1). \]Every vertex is transformed by the same operator.
Linear Transformations Fix the Origin
For any matrix \(A\),
\[ A\mathbf{0} = \mathbf{0}. \]Therefore, pure matrix transformations of the form
\[ \mathbf{v}\mapsto A\mathbf{v} \]always leave the origin fixed.
Why Ordinary Translations Need Special Treatment
A translation such as
\[ (x,y)\mapsto(x+a,y+b) \]generally moves the origin.
Therefore, it cannot be represented by an ordinary \(2\times2\) matrix acting alone on \(\begin{bmatrix}x\\y\end{bmatrix}\).
Translations are affine transformations rather than purely linear ones.
Combining Transformations
Several transformations can be applied in sequence.
For example, a vector may first be reflected and then rotated.
If
\[ \mathbf{v} \]is first transformed by \(A\) and then by \(B\), the result is
\[ \boxed{ B(A\mathbf{v}) }. \]Order Matters
In general,
\[ B(A\mathbf{v}) \neq A(B\mathbf{v}). \]Rotating and then reflecting can produce a different result from reflecting and then rotating.
Transformation composition is therefore order-sensitive.
Example: Reflect Then Rotate
Let
\[ \mathbf{v} = \begin{bmatrix} 2\\ 1 \end{bmatrix}. \]First reflect across the \(x\)-axis:
\[ \begin{bmatrix} 2\\ 1 \end{bmatrix} \mapsto \begin{bmatrix} 2\\ -1 \end{bmatrix}. \]Then rotate \(90^\circ\) counterclockwise:
\[ \begin{bmatrix} 2\\ -1 \end{bmatrix} \mapsto \boxed{ \begin{bmatrix} 1\\ 2 \end{bmatrix} }. \]Example: Rotate Then Reflect
Start again with
\[ \mathbf{v} = \begin{bmatrix} 2\\ 1 \end{bmatrix}. \]Rotate first:
\[ \begin{bmatrix} 2\\ 1 \end{bmatrix} \mapsto \begin{bmatrix} -1\\ 2 \end{bmatrix}. \]Then reflect across the \(x\)-axis:
\[ \begin{bmatrix} -1\\ 2 \end{bmatrix} \mapsto \boxed{ \begin{bmatrix} -1\\ -2 \end{bmatrix} }. \]The result differs from the previous order.
Transformation Invariants
An invariant is a property that remains unchanged under a transformation.
Different matrix families preserve different features.
| Transformation | Typical Preserved Features |
|---|---|
| Rotation | Lengths, angles, area, orientation |
| Reflection | Lengths, angles, area |
| Uniform scaling | Angles, shape, ratios of lengths |
| Nonuniform scaling | Parallelism, collinearity |
| Shear | Parallelism, collinearity, standard shear area |
Fixed Vectors and Fixed Directions
Some transformations leave certain vectors or directions unchanged.
Reflection across the \(x\)-axis fixes every vector lying on the \(x\)-axis.
A horizontal shear fixes the horizontal direction.
These ideas lead naturally toward eigenvectors in a later lesson.
Recognizing a Transformation from Its Matrix
Certain matrix patterns reveal familiar transformations.
For example,
\[ \begin{bmatrix} 0&-1\\ 1&0 \end{bmatrix} \]is a \(90^\circ\) rotation, while
\[ \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix} \]is reflection across the \(x\)-axis.
Reading a Transformation from Basis Images
Suppose
\[ A\mathbf{i} = \begin{bmatrix} 2\\ 0 \end{bmatrix} \]and
\[ A\mathbf{j} = \begin{bmatrix} 0\\ 3 \end{bmatrix}. \]Then
\[ A= \begin{bmatrix} 2&0\\ 0&3 \end{bmatrix}. \]This scales horizontally by \(2\) and vertically by \(3\).
Constructing a Matrix from Coordinate Rules
Suppose a transformation satisfies
\[ (x,y)\mapsto(2x+y,\;x-3y). \]Then
\[ \begin{bmatrix} x\\ y \end{bmatrix} \mapsto \begin{bmatrix} 2x+y\\ x-3y \end{bmatrix}. \]Therefore,
\[ \boxed{ A= \begin{bmatrix} 2&1\\ 1&-3 \end{bmatrix} }. \]Why Matrix Transformations Preserve the Origin and Lines through It
If a line through the origin consists of vectors
\[ t\mathbf{v}, \]then applying \(A\) gives
\[ A(t\mathbf{v}) = tA\mathbf{v}. \]The images still lie along a line through the origin, unless every vector collapses to the zero vector.
Common Error: Multiplying in the Wrong Pattern
For
\[ A= \begin{bmatrix} a&b\\ c&d \end{bmatrix} \]and
\[ \mathbf{v} = \begin{bmatrix} x\\ y \end{bmatrix}, \]the correct product is
\[ \boxed{ \begin{bmatrix} ax+by\\ cx+dy \end{bmatrix} }. \]Each matrix row combines with the entire vector column.
Common Error: Using the Clockwise Rotation Matrix for Counterclockwise Rotation
The standard counterclockwise rotation matrix is
\[ \boxed{ \begin{bmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta \end{bmatrix} }. \]Changing the signs in the wrong places reverses the direction of rotation.
Common Error: Reflecting the Wrong Coordinate
Reflection across the \(x\)-axis changes the sign of \(y\):
\[ (x,y)\mapsto(x,-y). \]Reflection across the \(y\)-axis changes the sign of \(x\):
\[ (x,y)\mapsto(-x,y). \]The coordinate perpendicular to the mirror line changes sign.
Common Error: Treating Nonuniform Scaling as Uniform
The matrix
\[ \begin{bmatrix} 2&0\\ 0&3 \end{bmatrix} \]does not enlarge every length by one common factor.
Horizontal and vertical directions scale differently, so angles and shapes may change.
Common Error: Confusing Horizontal and Vertical Shears
A horizontal shear changes the \(x\)-coordinate:
\[ (x,y)\mapsto(x+ky,y). \]A vertical shear changes the \(y\)-coordinate:
\[ (x,y)\mapsto(x,y+kx). \]The unchanged coordinate identifies the shear's primary direction.
Common Error: Assuming Composition Order Does Not Matter
Matrix transformations generally do not commute.
Applying \(A\) and then \(B\) can differ from applying \(B\) and then \(A\).
Always preserve the stated sequence of transformations.
Common Error: Trying to Represent Translation with a \(2\times2\) Matrix Alone
An ordinary \(2\times2\) linear transformation fixes the origin.
A nonzero translation moves the origin.
Therefore, translation requires an affine description or an expanded coordinate system such as homogeneous coordinates.
A Reliable Procedure for Applying a Transformation Matrix
- Write the vector as a column vector.
- Confirm that matrix and vector dimensions are compatible.
- Multiply each matrix row by the vector column.
- Simplify the resulting coordinates.
- Interpret the output geometrically.
- Check whether the result matches the expected direction, size, or symmetry.
A Reliable Procedure for Identifying a Standard Transformation
- Inspect the matrix entries.
- Determine where \(\mathbf{i}\) and \(\mathbf{j}\) are sent.
- Check whether lengths or directions appear preserved.
- Compare the pattern with standard rotation, reflection, scaling, or shear matrices.
- Test one simple vector if needed.
A Reliable Procedure for Transforming a Figure
- List the coordinates of every relevant vertex.
- Convert each vertex to a position vector.
- Apply the same matrix to every vector.
- Record the transformed coordinates.
- Reconnect corresponding image vertices.
- Compare preserved and changed geometric features.
A Reliable Procedure for Sequential Transformations
- Read the transformation order carefully.
- Apply the first transformation to the original vector.
- Use that output as the input to the second transformation.
- Continue in sequence.
- Do not reverse the order unless the problem explicitly asks you to compare orders.
- Check the final geometry against the expected combined effect.
Example: Full Rotation Workflow
Rotate
\[ \mathbf{v} = \begin{bmatrix} 4\\ -1 \end{bmatrix} \]counterclockwise by \(90^\circ\).
Use
\[ R_{90^\circ} = \begin{bmatrix} 0&-1\\ 1&0 \end{bmatrix}. \]Then
\[ R_{90^\circ}\mathbf{v} = \begin{bmatrix} 0(4)+(-1)(-1)\\ 1(4)+0(-1) \end{bmatrix}. \]Complete the Rotation Workflow
Simplify:
\[ \boxed{ R_{90^\circ}\mathbf{v} = \begin{bmatrix} 1\\ 4 \end{bmatrix} }. \]Check the magnitude:
\[ \sqrt{4^2+(-1)^2} = \sqrt{17}, \]while
\[ \sqrt{1^2+4^2} = \sqrt{17}. \]The rotation preserved length as expected.
Example: Full Shear Workflow
Apply the horizontal shear
\[ H= \begin{bmatrix} 1&3\\ 0&1 \end{bmatrix} \]to
\[ \mathbf{v} = \begin{bmatrix} -2\\ 4 \end{bmatrix}. \]Multiply:
\[ H\mathbf{v} = \begin{bmatrix} -2+3(4)\\ 4 \end{bmatrix}. \]Therefore,
\[ \boxed{ H\mathbf{v} = \begin{bmatrix} 10\\ 4 \end{bmatrix} }. \]Practice 1: Rotation by \(90^\circ\)
Rotate
\[ \begin{bmatrix} 2\\ 5 \end{bmatrix} \]counterclockwise by \(90^\circ\).
Practice 2: Rotation by \(180^\circ\)
Rotate
\[ \begin{bmatrix} -3\\ 4 \end{bmatrix} \]by \(180^\circ\).
Practice 3: Reflection across the \(x\)-Axis
Reflect
\[ \begin{bmatrix} 6\\ -2 \end{bmatrix} \]across the \(x\)-axis.
Practice 4: Reflection across \(y=x\)
Reflect
\[ \begin{bmatrix} 3\\ -7 \end{bmatrix} \]across the line \(y=x\).
Practice 5: Uniform Scaling
Apply
\[ S= \begin{bmatrix} 4&0\\ 0&4 \end{bmatrix} \]to
\[ \begin{bmatrix} -1\\ 3 \end{bmatrix}. \]Practice 6: Nonuniform Scaling
Apply
\[ S= \begin{bmatrix} 2&0\\ 0&\frac12 \end{bmatrix} \]to
\[ \begin{bmatrix} 4\\ 6 \end{bmatrix}. \]Practice 7: Horizontal Shear
Apply
\[ H= \begin{bmatrix} 1&-2\\ 0&1 \end{bmatrix} \]to
\[ \begin{bmatrix} 5\\ 3 \end{bmatrix}. \]Practice 8: Vertical Shear
Apply
\[ V= \begin{bmatrix} 1&0\\ 2&1 \end{bmatrix} \]to
\[ \begin{bmatrix} -2\\ 1 \end{bmatrix}. \]Practice 9: Identify the Transformation
Identify the transformation represented by
\[ \begin{bmatrix} -1&0\\ 0&1 \end{bmatrix}. \]Practice 10: Identify the Error
A student says that
\[ \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix} \]represents a translation \(3\) units right because it leaves vector lengths unchanged.
Explain the error.
Practice Answers
Practice 1
\[ \begin{bmatrix} 0&-1\\ 1&0 \end{bmatrix} \begin{bmatrix} 2\\ 5 \end{bmatrix} = \boxed{ \begin{bmatrix} -5\\ 2 \end{bmatrix} }. \]Practice 2
\[ \begin{bmatrix} -1&0\\ 0&-1 \end{bmatrix} \begin{bmatrix} -3\\ 4 \end{bmatrix} = \boxed{ \begin{bmatrix} 3\\ -4 \end{bmatrix} }. \]Practice 3
\[ \boxed{ \begin{bmatrix} 6\\ 2 \end{bmatrix} }. \]Practice 4
Reflection across \(y=x\) exchanges coordinates:
\[ \boxed{ \begin{bmatrix} -7\\ 3 \end{bmatrix} }. \]Practice 5
\[ \boxed{ \begin{bmatrix} -4\\ 12 \end{bmatrix} }. \]Practice 6
\[ \boxed{ \begin{bmatrix} 8\\ 3 \end{bmatrix} }. \]Practice 7
\[ \begin{bmatrix} 1&-2\\ 0&1 \end{bmatrix} \begin{bmatrix} 5\\ 3 \end{bmatrix} = \begin{bmatrix} 5-6\\ 3 \end{bmatrix} = \boxed{ \begin{bmatrix} -1\\ 3 \end{bmatrix} }. \]Practice 8
\[ \begin{bmatrix} 1&0\\ 2&1 \end{bmatrix} \begin{bmatrix} -2\\ 1 \end{bmatrix} = \begin{bmatrix} -2\\ -4+1 \end{bmatrix} = \boxed{ \begin{bmatrix} -2\\ -3 \end{bmatrix} }. \]Practice 9
The matrix changes the sign of \(x\) and leaves \(y\) unchanged.
Therefore, it represents
\[ \boxed{\text{reflection across the }y\text{-axis}}. \]Practice 10
The matrix is the identity matrix, so it sends every vector to itself.
It does not move the origin and therefore does not translate points.
A nonzero translation cannot be represented by an ordinary \(2\times2\) linear transformation matrix alone.
What to Remember
- Matrices can act as spatial transformation operators.
- The columns of a transformation matrix show the images of the standard basis vectors.
- Rotation through \(\theta\) uses the matrix \(\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}\).
- Rotations preserve lengths and angles.
- Reflections preserve lengths and angles but reverse orientation.
- Uniform scaling changes every direction by one common scale factor.
- Nonuniform scaling can distort angles and shapes.
- Horizontal and vertical shears shift one coordinate in proportion to the other.
- Standard shears preserve area and parallelism.
- Ordinary linear matrix transformations fix the origin.
- Transformation order matters when several matrices are applied in sequence.
Where We Go Next
Individual matrices can encode important geometric transformations. When several transformations are performed in sequence, we need a systematic way to combine their operators into one transformation.
The next lesson, Composition through Matrix Multiplication, develops matrix multiplication as the algebra of transformation composition and explains why the order of multiplication records the order in which transformations act.