The Central Idea
Let
\[ A= \begin{bmatrix} a&b\\ c&d \end{bmatrix} \]and let
\[ \mathbf{v} = \begin{bmatrix} x\\ y \end{bmatrix}. \]Matrix-vector multiplication gives
\[ \boxed{ A\mathbf{v} = \begin{bmatrix} ax+by\\ cx+dy \end{bmatrix} }. \]This product defines a transformation from input vectors to output vectors.
A Matrix Defines a Rule
We can think of \(A\) as a function:
\[ \boxed{ T(\mathbf{v})=A\mathbf{v} }. \]For each input vector \(\mathbf{v}\), the matrix produces exactly one output vector.
The matrix therefore acts as an operator on space.
Coordinate Rule from a Matrix
If
\[ A= \begin{bmatrix} a&b\\ c&d \end{bmatrix}, \]then the coordinate rule is
\[ \boxed{ (x,y) \mapsto (ax+by,\;cx+dy) }. \]Each output coordinate is a linear combination of the input coordinates.
Example: Read a Coordinate Rule
Let
\[ A= \begin{bmatrix} 2&1\\ -1&3 \end{bmatrix}. \]Then
\[ \begin{bmatrix} x\\ y \end{bmatrix} \mapsto \begin{bmatrix} 2x+y\\ -x+3y \end{bmatrix}. \]Therefore, the coordinate transformation is
\[ \boxed{ (x,y)\mapsto(2x+y,\;-x+3y) }. \]Example: Apply the Matrix
Using
\[ A= \begin{bmatrix} 2&1\\ -1&3 \end{bmatrix}, \]transform
\[ \mathbf{v} = \begin{bmatrix} 4\\ 2 \end{bmatrix}. \]Then
\[ A\mathbf{v} = \begin{bmatrix} 2(4)+1(2)\\ -1(4)+3(2) \end{bmatrix} = \boxed{ \begin{bmatrix} 10\\ 2 \end{bmatrix} }. \]Rows Determine Output Coordinates
In
\[ A= \begin{bmatrix} a&b\\ c&d \end{bmatrix}, \]the first row
\[ \begin{bmatrix} a&b \end{bmatrix} \]determines the first output coordinate:
\[ ax+by. \]The second row determines the second output coordinate:
\[ cx+dy. \]Columns Determine the Images of Basis Vectors
Recall the standard basis
\[ \mathbf{i} = \begin{bmatrix} 1\\ 0 \end{bmatrix}, \qquad \mathbf{j} = \begin{bmatrix} 0\\ 1 \end{bmatrix}. \]Then
\[ A\mathbf{i} = \begin{bmatrix} a\\ c \end{bmatrix} \]and
\[ A\mathbf{j} = \begin{bmatrix} b\\ d \end{bmatrix}. \]These are exactly the columns of \(A\).
Why the Columns Matter So Much
Every vector can be written as
\[ \mathbf{v} = x\mathbf{i}+y\mathbf{j}. \]A matrix transformation gives
\[ A\mathbf{v} = A(x\mathbf{i}+y\mathbf{j}). \]Linearity allows us to write
\[ A\mathbf{v} = xA\mathbf{i}+yA\mathbf{j}. \]Thus the transformation of every vector is determined by the images of the basis vectors.
Build a Matrix from Basis Images
Suppose a transformation sends
\[ \mathbf{i} \mapsto \begin{bmatrix} 2\\ 1 \end{bmatrix} \]and
\[ \mathbf{j} \mapsto \begin{bmatrix} -1\\ 3 \end{bmatrix}. \]Put these output vectors into the columns:
\[ \boxed{ A= \begin{bmatrix} 2&-1\\ 1&3 \end{bmatrix} }. \]Example: Use Basis Images to Transform a Vector
Let
\[ \mathbf{v} = 4\mathbf{i}+2\mathbf{j}. \]If
\[ A\mathbf{i} = \begin{bmatrix} 2\\ 1 \end{bmatrix} \]and
\[ A\mathbf{j} = \begin{bmatrix} -1\\ 3 \end{bmatrix}, \]then
\[ A\mathbf{v} = 4A\mathbf{i}+2A\mathbf{j}. \]Complete the Basis-Image Example
\[ A\mathbf{v} = 4 \begin{bmatrix} 2\\ 1 \end{bmatrix} + 2 \begin{bmatrix} -1\\ 3 \end{bmatrix}. \]Therefore,
\[ A\mathbf{v} = \begin{bmatrix} 8\\ 4 \end{bmatrix} + \begin{bmatrix} -2\\ 6 \end{bmatrix} = \boxed{ \begin{bmatrix} 6\\ 10 \end{bmatrix} }. \]What Makes the Transformation Linear?
A matrix transformation satisfies two key properties:
\[ \boxed{ A(\mathbf{u}+\mathbf{v}) = A\mathbf{u}+A\mathbf{v} } \]and
\[ \boxed{ A(c\mathbf{v}) = cA\mathbf{v} }. \]These properties preserve vector addition and scalar multiplication.
Additivity
Suppose
\[ \mathbf{u} = \begin{bmatrix} u_1\\ u_2 \end{bmatrix} \]and
\[ \mathbf{v} = \begin{bmatrix} v_1\\ v_2 \end{bmatrix}. \]Then
\[ A(\mathbf{u}+\mathbf{v}) \]produces the same result as transforming each vector separately and adding:
\[ \boxed{ A(\mathbf{u}+\mathbf{v}) = A\mathbf{u}+A\mathbf{v} }. \]Homogeneity
For a scalar \(c\),
\[ \boxed{ A(c\mathbf{v}) = c(A\mathbf{v}) }. \]Scaling before the transformation gives the same result as transforming first and scaling afterward.
Linearity and Linear Combinations
Combining additivity and homogeneity gives
\[ \boxed{ A(c_1\mathbf{v}_1+c_2\mathbf{v}_2) = c_1A\mathbf{v}_1+c_2A\mathbf{v}_2 }. \]More generally, a matrix acts term-by-term on any linear combination.
The Origin Is Always Fixed
For every matrix \(A\),
\[ A\mathbf{0} = \mathbf{0}. \]Therefore,
\[ \boxed{ \text{every linear matrix transformation fixes the origin} }. \]This fact distinguishes linear transformations from ordinary translations.
Why Translation Is Not Represented by a \(2\times2\) Matrix Alone
A translation such as
\[ (x,y)\mapsto(x+3,y-1) \]sends the origin to
\[ (3,-1). \]Since a linear matrix transformation must keep the origin fixed, a nonzero translation is not represented by an ordinary \(2\times2\) matrix acting alone on plane coordinate vectors.
Transforming a Line through the Origin
A line through the origin can be written
\[ \mathbf{r} = t\mathbf{v}. \]Applying \(A\) gives
\[ A\mathbf{r} = A(t\mathbf{v}) = tA\mathbf{v}. \]The image is therefore another line through the origin, unless \(A\mathbf{v}=\mathbf{0}\).
Transforming a General Line
A line may be written
\[ \mathbf{r} = \mathbf{r}_0+t\mathbf{v}. \]Applying \(A\):
\[ A\mathbf{r} = A\mathbf{r}_0+tA\mathbf{v}. \]Thus a line is usually sent to another line.
It may collapse to a point if its direction lies in the kernel.
Transforming a Plane
A plane through the origin spanned by
\[ \mathbf{u},\mathbf{v} \]consists of vectors
\[ s\mathbf{u}+t\mathbf{v}. \]Applying \(A\):
\[ A(s\mathbf{u}+t\mathbf{v}) = sA\mathbf{u}+tA\mathbf{v}. \]The transformed plane is generated by the transformed spanning vectors.
Transforming an Entire Figure
To transform a coordinate figure, apply the same matrix to every vertex.
If a polygon has vertices
\[ \mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n, \]then the image vertices are
\[ \boxed{ A\mathbf{v}_1,\; A\mathbf{v}_2,\; \ldots,\; A\mathbf{v}_n }. \]Example: Transform a Triangle
Let
\[ A= \begin{bmatrix} 2&0\\ 0&1 \end{bmatrix} \]and let a triangle have vertices
\[ P=(0,0),\qquad Q=(1,0),\qquad R=(0,1). \]Apply \(A\) to each position vector.
Complete the Triangle Transformation
\[ A \begin{bmatrix} 0\\ 0 \end{bmatrix} = \begin{bmatrix} 0\\ 0 \end{bmatrix}, \] \[ A \begin{bmatrix} 1\\ 0 \end{bmatrix} = \begin{bmatrix} 2\\ 0 \end{bmatrix}, \] \[ A \begin{bmatrix} 0\\ 1 \end{bmatrix} = \begin{bmatrix} 0\\ 1 \end{bmatrix}. \]The triangle is stretched horizontally by a factor of \(2\).
Some Matrices Stretch
The matrix
\[ \begin{bmatrix} 2&0\\ 0&3 \end{bmatrix} \]sends
\[ (x,y)\mapsto(2x,3y). \]It scales different coordinate directions by different amounts.
Some Matrices Reflect
The matrix
\[ \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix} \]sends
\[ (x,y)\mapsto(x,-y). \]This reflects the plane across the \(x\)-axis.
Some Matrices Rotate
The matrix
\[ \begin{bmatrix} 0&-1\\ 1&0 \end{bmatrix} \]sends
\[ (x,y)\mapsto(-y,x). \]This is a \(90^\circ\) counterclockwise rotation about the origin.
Some Matrices Shear
The matrix
\[ \begin{bmatrix} 1&k\\ 0&1 \end{bmatrix} \]sends
\[ (x,y)\mapsto(x+ky,y). \]This shifts horizontal position in proportion to vertical position.
Some Matrices Collapse Space
Consider
\[ A= \begin{bmatrix} 1&0\\ 0&0 \end{bmatrix}. \]Then
\[ (x,y)\mapsto(x,0). \]Every point is sent to the \(x\)-axis.
A two-dimensional plane has been collapsed into one dimension.
The Kernel
The kernel of a matrix transformation is the set of all input vectors sent to the zero vector.
\[ \boxed{ \ker(A) = \{\mathbf{v}:A\mathbf{v}=\mathbf{0}\} }. \]The kernel reveals directions that are completely collapsed.
Example: Kernel of a Projection-Like Matrix
Let
\[ A= \begin{bmatrix} 1&0\\ 0&0 \end{bmatrix}. \]Then
\[ A \begin{bmatrix} 0\\ y \end{bmatrix} = \begin{bmatrix} 0\\ 0 \end{bmatrix}. \]Therefore, the entire \(y\)-axis lies in the kernel.
The Image
The image or range of a matrix transformation is the set of all possible outputs:
\[ \boxed{ \operatorname{im}(A) = \{A\mathbf{v}:\mathbf{v}\text{ is an allowed input}\} }. \]The image describes the part of the output space actually reached by the transformation.
Columns Span the Image
Since
\[ A \begin{bmatrix} x\\ y \end{bmatrix} = xA\mathbf{i}+yA\mathbf{j}, \]every output is a linear combination of the columns of \(A\).
Therefore,
\[ \boxed{ \operatorname{im}(A) = \operatorname{span}\{\text{columns of }A\} }. \]Example: Image of a Collapsing Matrix
For
\[ A= \begin{bmatrix} 1&0\\ 0&0 \end{bmatrix}, \]the columns are
\[ \begin{bmatrix} 1\\ 0 \end{bmatrix} \]and
\[ \begin{bmatrix} 0\\ 0 \end{bmatrix}. \]Their span is the \(x\)-axis.
Thus the image is one-dimensional.
Invertible Transformations
A matrix transformation is invertible when every output comes from exactly one input.
If \(A\) is invertible, there exists a matrix
\[ A^{-1} \]such that
\[ \boxed{ A^{-1}A=AA^{-1}=I }. \]Geometric Meaning of Invertibility
An invertible transformation does not collapse distinct input vectors into the same output.
It preserves enough information for the original vector to be recovered.
Rotations, reflections, nonzero scalings, and standard shears are examples of invertible linear transformations.
Noninvertible Transformations
A transformation is noninvertible when information is lost.
For example,
\[ (x,y)\mapsto(x,0) \]erases the \(y\)-coordinate.
Once the output is known, the original \(y\)-value cannot be recovered.
Determinant as an Area-Scaling Signal
For a \(2\times2\) matrix
\[ A= \begin{bmatrix} a&b\\ c&d \end{bmatrix}, \]the determinant is
\[ \boxed{ \det(A)=ad-bc }. \]Its absolute value measures how area scales under the transformation.
Determinant and Invertibility
For a square matrix,
\[ \boxed{ \det(A)\neq0 } \]means the transformation is invertible.
If
\[ \boxed{ \det(A)=0, } \]the transformation collapses dimension and is not invertible.
Positive and Negative Determinants
In the plane:
- a positive determinant preserves orientation;
- a negative determinant reverses orientation;
- a zero determinant collapses area to zero.
Thus the determinant carries geometric information beyond simple area size.
Example: Area Scaling
Let
\[ A= \begin{bmatrix} 2&0\\ 0&3 \end{bmatrix}. \]Then
\[ \det(A)=6. \]Therefore, every planar area is multiplied by
\[ \boxed{6}. \]Example: Orientation Reversal
Reflection across the \(x\)-axis uses
\[ F= \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix}. \]Its determinant is
\[ -1. \]The absolute area is unchanged, but orientation is reversed.
Three-Dimensional Matrix Transformations
A \(3\times3\) matrix can act on a three-dimensional vector:
\[ A \begin{bmatrix} x\\ y\\ z \end{bmatrix}. \]The output has three coordinates.
The columns of \(A\) show where the standard basis vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) are sent.
General Dimension Pattern
An
\[ m\times n \]matrix acts naturally on an \(n\)-component column vector and produces an \(m\)-component output vector.
Symbolically,
\[ \boxed{ A:\mathbb{R}^n\to\mathbb{R}^m }. \]Square versus Rectangular Transformation Matrices
A square matrix maps a coordinate space back into a space of the same dimension.
A rectangular matrix can map between spaces of different dimensions.
For example, a \(2\times3\) matrix maps three-component inputs to two-component outputs.
Example: Mapping from \(\mathbb{R}^3\) to \(\mathbb{R}^2\)
Let
\[ A= \begin{bmatrix} 1&0&2\\ 0&1&-1 \end{bmatrix}. \]Then
\[ A \begin{bmatrix} x\\ y\\ z \end{bmatrix} = \begin{bmatrix} x+2z\\ y-z \end{bmatrix}. \]The transformation converts a 3D vector into a 2D vector.
Fixed Directions
Some transformations leave particular directions unchanged or only scale them.
These directions satisfy equations of the form
\[ A\mathbf{v} = \lambda\mathbf{v}. \]They will later be studied as eigenvector directions.
Why Transformation Matrices Are Powerful
Matrix operators let us:
- encode geometric transformations compactly;
- apply the same rule to many vectors;
- analyze how basis directions change;
- study lines, planes, and higher-dimensional subspaces;
- detect collapsed directions through the kernel;
- describe reachable outputs through the image;
- combine transformations by matrix multiplication;
- change coordinate systems systematically.
Common Error: Treating a Matrix as Only a Data Table
A matrix can store data, but in this context it represents an operation.
The entries determine how input coordinates are combined to produce output coordinates.
Common Error: Multiplying Component by Component
For
\[ A= \begin{bmatrix} a&b\\ c&d \end{bmatrix} \]and
\[ \mathbf{v} = \begin{bmatrix} x\\ y \end{bmatrix}, \]the product is
\[ \boxed{ \begin{bmatrix} ax+by\\ cx+dy \end{bmatrix} }, \]not a list of entry-by-entry products.
Common Error: Confusing Rows and Columns
Rows determine output coordinate formulas.
Columns show the images of the input basis vectors.
Both interpretations are useful, but they answer different questions.
Common Error: Forgetting That Linear Transformations Fix the Origin
Every matrix transformation satisfies
\[ A\mathbf{0}=\mathbf{0}. \]If a proposed coordinate rule moves the origin, it is not represented by an ordinary linear matrix transformation alone.
Common Error: Assuming Every Matrix Is Invertible
Some matrices collapse dimensions.
If nonzero vectors lie in the kernel, the transformation loses information.
Such a matrix cannot have an inverse.
Common Error: Assuming Every Transformation Preserves Length
Rotations and reflections preserve length, but general matrices need not.
Scalings and shears can change magnitudes, angles, and shapes.
Always identify what the specific matrix preserves.
Common Error: Assuming Every Transformation Preserves Dimension
A matrix can collapse a plane to a line or a line to a point.
Dimension preservation is tied to independence, rank, and invertibility.
A Reliable Procedure for Applying a Matrix Operator
- Write the input as a column vector.
- Check that the matrix dimensions are compatible with the vector.
- Multiply each matrix row by the vector column.
- Simplify each output coordinate.
- Interpret the result geometrically when useful.
- Check whether the transformation behaves as expected.
A Reliable Procedure for Building a Matrix from a Coordinate Rule
- Write each output coordinate as a linear expression in the inputs.
- Read the coefficients of \(x\), \(y\), and any additional variables.
- Place the coefficients for the first output coordinate in the first row.
- Place the coefficients for the second output coordinate in the second row.
- Continue for higher dimensions.
- Verify by multiplying a general input vector.
A Reliable Procedure for Building a Matrix from Basis Images
- Identify the image of each standard basis vector.
- Place the image of the first basis vector in the first column.
- Place the image of the second basis vector in the second column.
- Continue for all basis vectors.
- Test the matrix on a general linear combination.
A Reliable Procedure for Analyzing a Transformation Geometrically
- Inspect the images of the basis vectors.
- Check whether the origin is fixed.
- Look for preserved or collapsed directions.
- Examine how lengths, angles, and area change.
- Check whether the transformation is invertible.
- Use the determinant and kernel when appropriate.
Example: Full Operator Workflow
Let
\[ A= \begin{bmatrix} 1&2\\ 0&3 \end{bmatrix} \]and
\[ \mathbf{v} = \begin{bmatrix} 2\\ -1 \end{bmatrix}. \]Compute
\[ A\mathbf{v}. \]Complete the Full Operator Workflow
\[ A\mathbf{v} = \begin{bmatrix} 1(2)+2(-1)\\ 0(2)+3(-1) \end{bmatrix}. \]Therefore,
\[ \boxed{ A\mathbf{v} = \begin{bmatrix} 0\\ -3 \end{bmatrix} }. \]The input vector is transformed into a vector pointing straight downward.
Example: Build a Matrix from a Rule
Suppose
\[ T(x,y) = (3x-2y,\;x+4y). \]The coefficients of the first output coordinate are \(3\) and \(-2\).
The coefficients of the second output coordinate are \(1\) and \(4\).
Therefore,
\[ \boxed{ A= \begin{bmatrix} 3&-2\\ 1&4 \end{bmatrix} }. \]Example: Build a Matrix from Basis Images
Suppose
\[ T(\mathbf{i}) = \begin{bmatrix} 0\\ 2 \end{bmatrix} \]and
\[ T(\mathbf{j}) = \begin{bmatrix} -1\\ 1 \end{bmatrix}. \]Therefore,
\[ \boxed{ A= \begin{bmatrix} 0&-1\\ 2&1 \end{bmatrix} }. \]Practice 1: Apply a Matrix
Let
\[ A= \begin{bmatrix} 2&1\\ 0&3 \end{bmatrix}. \]Find
\[ A \begin{bmatrix} 4\\ 2 \end{bmatrix}. \]Practice 2: Read a Coordinate Rule
Write the coordinate rule represented by
\[ A= \begin{bmatrix} 1&-2\\ 3&4 \end{bmatrix}. \]Practice 3: Build a Matrix from a Rule
Find the matrix for
\[ T(x,y) = (2x+5y,\;-x+y). \]Practice 4: Read Basis Images
Let
\[ A= \begin{bmatrix} 3&-1\\ 2&4 \end{bmatrix}. \]Find \(A\mathbf{i}\) and \(A\mathbf{j}\).
Practice 5: Build a Matrix from Basis Images
A transformation sends
\[ \mathbf{i} \mapsto \begin{bmatrix} 2\\ -1 \end{bmatrix} \]and
\[ \mathbf{j} \mapsto \begin{bmatrix} 5\\ 3 \end{bmatrix}. \]Find its matrix.
Practice 6: Kernel Recognition
Let
\[ A= \begin{bmatrix} 1&0\\ 0&0 \end{bmatrix}. \]Describe the kernel geometrically.
Practice 7: Image Recognition
For the same matrix
\[ A= \begin{bmatrix} 1&0\\ 0&0 \end{bmatrix}, \]describe the image geometrically.
Practice 8: Determinant and Area
Let
\[ A= \begin{bmatrix} 2&0\\ 0&5 \end{bmatrix}. \]By what factor does the transformation scale area?
Practice 9: Invertibility
Is
\[ A= \begin{bmatrix} 1&2\\ 2&4 \end{bmatrix} \]invertible?
Practice 10: Identify the Error
A student claims that the rule
\[ (x,y)\mapsto(x+2,y) \]can be represented by a \(2\times2\) matrix because it is a geometric transformation.
Explain the error.
Practice Answers
Practice 1
\[ \begin{bmatrix} 2&1\\ 0&3 \end{bmatrix} \begin{bmatrix} 4\\ 2 \end{bmatrix} = \begin{bmatrix} 10\\ 6 \end{bmatrix}. \]Therefore,
\[ \boxed{ \begin{bmatrix} 10\\ 6 \end{bmatrix} }. \]Practice 2
\[ \boxed{ (x,y) \mapsto (x-2y,\;3x+4y) }. \]Practice 3
\[ \boxed{ A= \begin{bmatrix} 2&5\\ -1&1 \end{bmatrix} }. \]Practice 4
The first column is \(A\mathbf{i}\):
\[ \boxed{ A\mathbf{i} = \begin{bmatrix} 3\\ 2 \end{bmatrix} }. \]The second column is \(A\mathbf{j}\):
\[ \boxed{ A\mathbf{j} = \begin{bmatrix} -1\\ 4 \end{bmatrix} }. \]Practice 5
\[ \boxed{ A= \begin{bmatrix} 2&5\\ -1&3 \end{bmatrix} }. \]Practice 6
The kernel is the \(y\)-axis:
\[ \boxed{ \left\{ \begin{bmatrix} 0\\ y \end{bmatrix} :y\in\mathbb{R} \right\} }. \]Practice 7
The image is the \(x\)-axis:
\[ \boxed{ \left\{ \begin{bmatrix} x\\ 0 \end{bmatrix} :x\in\mathbb{R} \right\} }. \]Practice 8
\[ \det(A) = 2(5) = 10. \]Therefore, area is scaled by
\[ \boxed{10}. \]Practice 9
\[ \det(A) = 1(4)-2(2) = 0. \]Therefore, the matrix is not invertible.
Practice 10
The rule is a translation \(2\) units to the right.
It sends the origin to
\[ (2,0), \]while every ordinary \(2\times2\) linear transformation matrix must fix the origin.
Therefore, the translation cannot be represented by a \(2\times2\) matrix alone.
What to Remember
- A matrix can act as a transformation operator on vectors.
- Matrix-vector multiplication produces the output coordinates.
- Rows determine output coordinate formulas.
- Columns show where the standard basis vectors are sent.
- A matrix transformation preserves vector addition and scalar multiplication.
- Every linear matrix transformation fixes the origin.
- The kernel contains vectors collapsed to zero.
- The image is spanned by the columns of the matrix.
- Invertible transformations preserve enough information to recover inputs.
- The determinant records signed area scaling in two dimensions.
- Transformation matrices can stretch, reflect, rotate, shear, or collapse space.
Where We Go Next
A general matrix can act as a linear transformation, but certain matrices correspond to especially important geometric motions and deformations.
The next lesson, Rotation, Reflection, Scaling, and Shear Matrices, develops the standard matrices for these transformations and examines exactly what each one preserves and changes.