The Central Idea
Suppose
\[ T:\mathbb{R}^n\to\mathbb{R}^m \]is linear.
Because every vector in \(\mathbb{R}^n\) is a linear combination of the standard basis vectors \(e_1,\ldots,e_n\), the entire transformation is determined by
\[ T(e_1),\ldots,T(e_n). \]Place those output vectors into the columns of a matrix:
\[ \boxed{ A= \begin{bmatrix} \vert & \vert & & \vert\\ T(e_1) & T(e_2) & \cdots & T(e_n)\\ \vert & \vert & & \vert \end{bmatrix}}. \]Then
\[ \boxed{T(x)=Ax}. \]The Standard Matrix of a Linear Transformation
The matrix
\[ A= \begin{bmatrix} T(e_1)&T(e_2)&\cdots&T(e_n) \end{bmatrix} \]is called the standard matrix of the linear transformation \(T:\mathbb{R}^n\to\mathbb{R}^m\).
It contains \(m\) rows and \(n\) columns:
\[ \boxed{A\in\mathbb{R}^{m\times n}}. \]Why the Columns Are Basis Images
Any vector
\[ x= \begin{bmatrix} x_1\\ \vdots\\ x_n \end{bmatrix} \]can be written as
\[ x=x_1e_1+\cdots+x_ne_n. \]By linearity,
\[ T(x) = x_1T(e_1)+\cdots+x_nT(e_n). \]Matrix multiplication computes exactly this same linear combination of the columns.
Matrix Multiplication as a Column Combination
If
\[ A= \begin{bmatrix} \vert & \vert & & \vert\\ a_1&a_2&\cdots&a_n\\ \vert & \vert & & \vert \end{bmatrix} \]and
\[ x= \begin{bmatrix} x_1\\ \vdots\\ x_n \end{bmatrix}, \]then
\[ \boxed{Ax=x_1a_1+\cdots+x_na_n}. \]This is the key conceptual interpretation of matrix-vector multiplication.
Example: Build the Standard Matrix
Let
\[ T(x,y)=(2x-y,\;3x+4y). \]Evaluate the standard basis vectors:
\[ T(e_1)=T(1,0)= \begin{bmatrix} 2\\ 3 \end{bmatrix}, \] \[ T(e_2)=T(0,1)= \begin{bmatrix} -1\\ 4 \end{bmatrix}. \]Therefore the standard matrix is
\[ \boxed{ A= \begin{bmatrix} 2&-1\\ 3&4 \end{bmatrix}}. \]Check the Matrix Representation
Using
\[ A= \begin{bmatrix} 2&-1\\ 3&4 \end{bmatrix}, \]multiply by
\[ \begin{bmatrix} x\\ y \end{bmatrix}. \]We obtain
\[ A \begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} 2x-y\\ 3x+4y \end{bmatrix}. \]This reproduces the original transformation rule.
From a Matrix to a Linear Transformation
Every \(m\times n\) matrix defines a linear transformation
\[ T:\mathbb{R}^n\to\mathbb{R}^m \]by
\[ \boxed{T(x)=Ax}. \]Thus matrices and finite-dimensional coordinate linear transformations are two descriptions of the same structure.
Example: Read the Transformation from a Matrix
Let
\[ A= \begin{bmatrix} 1&2\\ -3&4 \end{bmatrix}. \]Then
\[ T(x,y) = A \begin{bmatrix} x\\ y \end{bmatrix} = \boxed{(x+2y,\;-3x+4y)}. \]Matrix Dimensions Encode Domain and Codomain
If
\[ A\in\mathbb{R}^{m\times n}, \]then \(A\) defines a map
\[ \boxed{\mathbb{R}^n\to\mathbb{R}^m}. \]The number of columns equals the input dimension.
The number of rows equals the output coordinate dimension.
Example: A \(3\times2\) Matrix
Let
\[ A= \begin{bmatrix} 1&0\\ 0&1\\ 1&1 \end{bmatrix}. \]This matrix defines
\[ T:\mathbb{R}^2\to\mathbb{R}^3. \]Specifically,
\[ T(x,y) = \begin{bmatrix} x\\ y\\ x+y \end{bmatrix}. \]Columns Describe Output Directions
For the matrix
\[ A= \begin{bmatrix} 1&0\\ 0&1\\ 1&1 \end{bmatrix}, \]the columns are
\[ a_1= \begin{bmatrix} 1\\ 0\\ 1 \end{bmatrix}, \qquad a_2= \begin{bmatrix} 0\\ 1\\ 1 \end{bmatrix}. \]Therefore,
\[ T(x,y)=xa_1+ya_2. \]The image of \(T\) is the span of these columns.
Column Space and Image
For a matrix transformation
\[ T(x)=Ax, \]the image of \(T\) is the column space of \(A\):
\[ \boxed{\operatorname{Im}(T)=\operatorname{Col}(A)}. \]Every possible output is a linear combination of the matrix columns.
Null Space and Kernel
The kernel of the matrix transformation satisfies
\[ Ax=0. \]Therefore,
\[ \boxed{\ker(T)=\operatorname{Null}(A)}. \]The null space records the input directions that the matrix sends to zero.
Rank of a Matrix Representation
Since the image equals the column space,
\[ \operatorname{rank}(T) = \dim(\operatorname{Im}(T)) = \dim(\operatorname{Col}(A)). \]Thus
\[ \boxed{\operatorname{rank}(T)=\operatorname{rank}(A)}. \]Nullity of a Matrix Representation
Since the kernel equals the null space,
\[ \operatorname{nullity}(T) = \dim(\operatorname{Null}(A)). \]Rank-nullity becomes
\[ \boxed{ n= \operatorname{rank}(A) + \operatorname{nullity}(A)} \]for an \(m\times n\) matrix.
The Identity Matrix
The identity transformation
\[ I(x)=x \]is represented by the identity matrix
\[ I_n= \begin{bmatrix} 1&0&\cdots&0\\ 0&1&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&1 \end{bmatrix}. \]Then
\[ I_nx=x. \]The Zero Matrix
The zero transformation is represented by the zero matrix.
For every input vector \(x\),
\[ 0x=0. \]Its image has dimension \(0\), while its null space is the entire domain.
Scaling Matrices
Uniform scaling in \(\mathbb{R}^2\),
\[ T(x,y)=(cx,cy), \]is represented by
\[ \boxed{ \begin{bmatrix} c&0\\ 0&c \end{bmatrix}}. \]More generally, diagonal matrices scale coordinate directions independently.
Reflection Matrices
Reflection across the \(x\)-axis is represented by
\[ \boxed{ \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix}}. \]Reflection across the \(y\)-axis is represented by
\[ \boxed{ \begin{bmatrix} -1&0\\ 0&1 \end{bmatrix}}. \]Rotation Matrices
Rotation through angle \(\theta\) about the origin is represented by
\[ \boxed{ R_\theta= \begin{bmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta \end{bmatrix}}. \]Its columns are the rotated images of the standard basis vectors.
Shear Matrices
The shear
\[ T(x,y)=(x+ky,y) \]is represented by
\[ \boxed{ \begin{bmatrix} 1&k\\ 0&1 \end{bmatrix}}. \]The first basis vector remains fixed, while the second acquires a horizontal component.
Projection Matrices
Projection of \(\mathbb{R}^3\) onto the \(xy\)-plane is represented by
\[ \boxed{ P= \begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&0 \end{bmatrix}}. \]The third standard basis vector is sent to zero.
A Matrix Representation Depends on Bases
The standard matrix assumes standard coordinate bases.
More generally, let
\[ B=(v_1,\ldots,v_n) \]be a basis for the domain and
\[ C=(w_1,\ldots,w_m) \]a basis for the codomain.
Then \(T\) has a matrix representation relative to \(B\) and \(C\).
Matrix Relative to Two Bases
For each domain basis vector \(v_j\), compute \(T(v_j)\) and express that output in the codomain basis \(C\).
If
\[ [T(v_j)]_C \]is its coordinate vector, place those coordinate vectors into the columns:
\[ \boxed{ [T]_{C\leftarrow B} = \begin{bmatrix} [T(v_1)]_C& \cdots& [T(v_n)]_C \end{bmatrix}}. \]The Basis-Coordinate Rule
If \(A=[T]_{C\leftarrow B}\), then
\[ \boxed{ [T(v)]_C = A[v]_B}. \]This equation is the general basis-dependent version of
\[ T(x)=Ax. \]Example: Nonstandard Domain Basis
Let
\[ B= \left( \begin{bmatrix}1\\1\end{bmatrix}, \begin{bmatrix}1\\-1\end{bmatrix} \right) \]and define
\[ T(x,y)=(2x,y). \]First compute
\[ T \left( \begin{bmatrix}1\\1\end{bmatrix} \right) = \begin{bmatrix} 2\\ 1 \end{bmatrix}, \] \[ T \left( \begin{bmatrix}1\\-1\end{bmatrix} \right) = \begin{bmatrix} 2\\ -1 \end{bmatrix}. \]If the codomain uses the standard basis, these outputs become the columns of the representation.
Ordered Bases Matter
A basis used for coordinates must be ordered.
Swapping
\[ B=(v_1,v_2) \]to
\[ B'=(v_2,v_1) \]changes coordinate columns and therefore changes the matrix representation.
The underlying transformation itself does not change.
Same Transformation, Different Matrices
A linear transformation is an abstract map.
A matrix is a coordinate representation of that map relative to chosen bases.
Different bases can therefore produce different matrices for the same transformation.
Change-of-Basis Matrices
Suppose \(B=(v_1,\ldots,v_n)\) is a basis for \(\mathbb{R}^n\).
Form the matrix
\[ P_B= \begin{bmatrix} \vert&&\vert\\ v_1&\cdots&v_n\\ \vert&&\vert \end{bmatrix}. \]Then
\[ \boxed{x=P_B[x]_B}. \]The matrix \(P_B\) converts basis coordinates into standard coordinates.
Converting Standard Coordinates to Basis Coordinates
If \(P_B\) is invertible, then
\[ x=P_B[x]_B \]implies
\[ \boxed{[x]_B=P_B^{-1}x}. \]Thus the inverse change-of-basis matrix converts standard coordinates into \(B\)-coordinates.
Changing the Matrix Representation
If a transformation \(T:\mathbb{R}^n\to\mathbb{R}^m\) has standard matrix \(A\), domain basis \(B\), and codomain basis \(C\), then
\[ \boxed{ [T]_{C\leftarrow B} = P_C^{-1}AP_B}. \]This formula says:
- convert \(B\)-coordinates to standard coordinates;
- apply the standard matrix \(A\);
- convert the result into \(C\)-coordinates.
Composition Becomes Matrix Multiplication
Suppose
\[ T:\mathbb{R}^n\to\mathbb{R}^m \]has matrix \(A\), and
\[ S:\mathbb{R}^m\to\mathbb{R}^p \]has matrix \(B\).
Then
\[ (S\circ T)(x)=B(Ax). \]Therefore,
\[ \boxed{[S\circ T]=BA}. \]Why Matrix Multiplication Order Matters
In
\[ BAx, \]the matrix \(A\) acts first and \(B\) acts second.
This mirrors function composition:
\[ S\circ T. \]Matrix multiplication is therefore generally not commutative.
Example: Compose Two Transformations
Let
\[ A= \begin{bmatrix} 2&0\\ 0&1 \end{bmatrix} \]represent horizontal stretching, and let
\[ B= \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix} \]represent reflection across the \(x\)-axis.
Applying \(A\) first and then \(B\) gives
\[ BA= \begin{bmatrix} 2&0\\ 0&-1 \end{bmatrix}. \]Inverse Transformations and Inverse Matrices
If a square linear transformation \(T\) is invertible and has matrix \(A\), then
\[ T^{-1} \]is represented by
\[ \boxed{A^{-1}}. \]The inverse identities are
\[ A^{-1}A=I \]and
\[ AA^{-1}=I. \]Invertibility Requires a Square Representation
A linear transformation between finite-dimensional spaces can be invertible only when the domain and codomain have the same dimension.
Therefore a matrix representing an invertible coordinate transformation must be square.
A Square Matrix Is Not Automatically Invertible
Consider
\[ A= \begin{bmatrix} 1&2\\ 2&4 \end{bmatrix}. \]The second column is twice the first.
The columns are dependent, so the transformation collapses two-dimensional input into a one-dimensional image.
Therefore \(A\) is not invertible.
Invertibility and Basis Images
A linear map
\[ T:\mathbb{R}^n\to\mathbb{R}^n \]is invertible exactly when the images
\[ T(e_1),\ldots,T(e_n) \]form a basis for \(\mathbb{R}^n\).
Equivalently, the columns of the standard matrix must be linearly independent and span the codomain.
Rows and Output Coordinates
Although columns describe where input basis vectors go, each row describes how one output coordinate depends on the input coordinates.
For
\[ A= \begin{bmatrix} 2&-1\\ 3&4 \end{bmatrix}, \]the first output coordinate is
\[ 2x-y, \]while the second is
\[ 3x+4y. \]Column View versus Row View
Two complementary interpretations are useful:
- column view: the input coordinates weight output direction vectors;
- row view: each output coordinate is a linear combination of input coordinates.
Both interpretations describe the same multiplication \(Ax\).
Matrix Representation of a Polynomial Transformation
Consider differentiation
\[ D:P_2\to P_1. \]Use bases
\[ B=(1,x,x^2) \]and
\[ C=(1,x). \]Compute
\[ D(1)=0,\qquad D(x)=1,\qquad D(x^2)=2x. \]Their \(C\)-coordinate vectors are
\[ \begin{bmatrix}0\\0\end{bmatrix}, \qquad \begin{bmatrix}1\\0\end{bmatrix}, \qquad \begin{bmatrix}0\\2\end{bmatrix}. \]Therefore,
\[ \boxed{ [D]_{C\leftarrow B} = \begin{bmatrix} 0&1&0\\ 0&0&2 \end{bmatrix}}. \]Use the Polynomial Matrix
Let
\[ p(x)=a+bx+cx^2. \]Its \(B\)-coordinate vector is
\[ [p]_B= \begin{bmatrix} a\\ b\\ c \end{bmatrix}. \]Then
\[ [D(p)]_C = \begin{bmatrix} 0&1&0\\ 0&0&2 \end{bmatrix} \begin{bmatrix} a\\ b\\ c \end{bmatrix} = \begin{bmatrix} b\\ 2c \end{bmatrix}. \]This corresponds to
\[ D(p)=b+2cx. \]Matrix Representation of Evaluation
Define
\[ T:P_2\to\mathbb{R} \]by
\[ T(p)=p(1). \]Relative to \(B=(1,x,x^2)\),
\[ T(1)=1,\qquad T(x)=1,\qquad T(x^2)=1. \]Therefore,
\[ \boxed{ [T]_B= \begin{bmatrix} 1&1&1 \end{bmatrix}}. \]Matrix Representation of a Matrix-Space Transformation
Linear transformations can also act on matrices.
For example, define
\[ T:M_{2\times2}\to\mathbb{R} \]by
\[ T(A)=\operatorname{tr}(A), \]the trace.
Once a basis for \(M_{2\times2}\) is chosen, this transformation also has a matrix representation.
Representations Turn Abstract Maps into Data
A matrix records a finite list of coefficients.
Those coefficients are enough to reconstruct the action of a linear transformation on every vector in the domain.
Matrix representation therefore converts an abstract structural rule into a computational object.
Common Error: Putting Basis Images in Rows
For the standard matrix, the images
\[ T(e_1),T(e_2),\ldots,T(e_n) \]go into the columns, not the rows.
This follows directly from the column-combination interpretation of matrix-vector multiplication.
Common Error: Reversing Matrix Dimensions
A map
\[ T:\mathbb{R}^n\to\mathbb{R}^m \]requires an
\[ \boxed{m\times n} \]matrix.
Inputs have \(n\) coordinates, so the matrix needs \(n\) columns. Outputs have \(m\) coordinates, so the matrix needs \(m\) rows.
Common Error: Confusing the Matrix with the Transformation
The transformation is the underlying linear map.
The matrix is its representation relative to specified bases.
Changing bases can change the matrix without changing the transformation.
Common Error: Ignoring Basis Order
If the basis is
\[ B=(v_1,v_2,v_3), \]then the first matrix column corresponds to \(v_1\), the second to \(v_2\), and the third to \(v_3\).
Reordering the basis changes the representation.
Common Error: Multiplying Composition Matrices Backward
If \(T\) has matrix \(A\) and \(S\) has matrix \(B\), then
\[ S\circ T \]is represented by
\[ \boxed{BA}, \]not \(AB\).
Common Error: Assuming Every Rectangular Matrix Is Invertible
An inverse matrix in the ordinary two-sided sense requires a square matrix.
Rectangular matrices can still be one-to-one or onto in certain dimension configurations, but they cannot possess a standard square inverse.
Common Error: Reading Rank from Matrix Size
An \(m\times n\) matrix has rank at most
\[ \min(m,n), \]but its actual rank depends on linear independence among its columns or rows.
Matrix size places an upper bound on rank; it does not determine rank.
A Reliable Procedure for Building a Standard Matrix
- Identify the domain \(\mathbb{R}^n\) and codomain \(\mathbb{R}^m\).
- List the standard basis vectors \(e_1,\ldots,e_n\).
- Compute \(T(e_1),\ldots,T(e_n)\).
- Place the outputs into matrix columns in the same order.
- Check that the matrix has \(m\) rows and \(n\) columns.
- Verify by computing \(Ax\) and comparing with \(T(x)\).
A Reliable Procedure for a Nonstandard Representation
- Identify the ordered domain basis \(B=(v_1,\ldots,v_n)\).
- Identify the ordered codomain basis \(C=(w_1,\ldots,w_m)\).
- Compute each \(T(v_j)\).
- Express each output in \(C\)-coordinates.
- Place \([T(v_j)]_C\) into column \(j\).
- Verify using \([T(v)]_C=A[v]_B\).
A Reliable Procedure for Reading a Transformation from a Matrix
- Identify the input coordinate vector \(x\).
- Compute \(Ax\).
- Read each row as one output coordinate formula.
- Interpret each column as the image of one input basis vector.
- Use the column span to understand the image.
- Use \(Ax=0\) to investigate the kernel.
Example: Full Standard-Matrix Workflow
Let
\[ T:\mathbb{R}^3\to\mathbb{R}^2 \]be defined by
\[ T(x,y,z)=(x+2y-z,\;3x+y+4z). \]Evaluate the standard basis:
\[ T(e_1)= \begin{bmatrix} 1\\ 3 \end{bmatrix}, \] \[ T(e_2)= \begin{bmatrix} 2\\ 1 \end{bmatrix}, \] \[ T(e_3)= \begin{bmatrix} -1\\ 4 \end{bmatrix}. \]Therefore,
\[ \boxed{ A= \begin{bmatrix} 1&2&-1\\ 3&1&4 \end{bmatrix}}. \]Indeed,
\[ A \begin{bmatrix} x\\ y\\ z \end{bmatrix} = \begin{bmatrix} x+2y-z\\ 3x+y+4z \end{bmatrix}. \]Example: Full Basis-Representation Workflow
Let
\[ T:\mathbb{R}^2\to\mathbb{R}^2 \]be defined by
\[ T(x,y)=(x+y,\;x-y). \]Use domain basis
\[ B= \left( \begin{bmatrix}1\\1\end{bmatrix}, \begin{bmatrix}1\\-1\end{bmatrix} \right) \]and standard codomain basis.
Compute
\[ T \left( \begin{bmatrix} 1\\ 1 \end{bmatrix} \right) = \begin{bmatrix} 2\\ 0 \end{bmatrix}, \] \[ T \left( \begin{bmatrix} 1\\ -1 \end{bmatrix} \right) = \begin{bmatrix} 0\\ 2 \end{bmatrix}. \]Therefore,
\[ \boxed{ [T]_{\text{std}\leftarrow B} = \begin{bmatrix} 2&0\\ 0&2 \end{bmatrix}}. \]Relative to this domain basis, the transformation becomes especially simple.
Why Matrix Representations Matter in Multidimensional Systems
Matrix representations are the computational language of finite-dimensional linear systems.
They allow transformations to be stored, composed, inverted, solved, approximated, and analyzed using systematic numerical procedures.
In applications, matrices can encode coordinate changes, networks, physical couplings, state transitions, feature mappings, constraints, measurement systems, and data transformations.
Practice 1: Build a Standard Matrix
Find the standard matrix of
\[ T(x,y)=(3x+2y,\;x-y). \]Practice 2: Read a Transformation
Let
\[ A= \begin{bmatrix} 2&-1\\ 4&3 \end{bmatrix}. \]Write the corresponding transformation \(T(x,y)\).
Practice 3: Determine Domain and Codomain
A matrix has size \(4\times3\). What coordinate-space transformation does it define?
Practice 4: Compute an Output
Let
\[ A= \begin{bmatrix} 1&2\\ -1&3 \end{bmatrix}. \]Compute
\[ A \begin{bmatrix} 4\\ -2 \end{bmatrix}. \]Practice 5: Column Interpretation
For
\[ A= \begin{bmatrix} 1&0\\ 2&3\\ 0&-1 \end{bmatrix}, \]identify \(T(e_1)\) and \(T(e_2)\).
Practice 6: Image from Columns
Describe the image of the transformation represented by
\[ A= \begin{bmatrix} 1&2\\ 2&4 \end{bmatrix}. \]Practice 7: Composition
Let
\[ A= \begin{bmatrix} 2&0\\ 0&1 \end{bmatrix} \]represent \(T\), and let
\[ B= \begin{bmatrix} 0&-1\\ 1&0 \end{bmatrix} \]represent \(S\).
Which product represents \(S\circ T\)?
Practice 8: Polynomial Representation
For \(D:P_2\to P_1\) with bases
\[ B=(1,x,x^2),\qquad C=(1,x), \]write the matrix representing differentiation.
Practice 9: Identity Matrix
What matrix represents the identity transformation on \(\mathbb{R}^3\)?
Practice 10: Invertibility Reasoning
Is
\[ A= \begin{bmatrix} 1&2\\ 2&4 \end{bmatrix} \]invertible? Explain using its columns.
Practice 11: Change of Coordinates
If
\[ P_B= \begin{bmatrix} 1&1\\ 1&-1 \end{bmatrix} \]and
\[ [x]_B= \begin{bmatrix} 3\\ 2 \end{bmatrix}, \]find \(x\) in standard coordinates.
Practice 12: Matrix Size and Rank
What is the largest possible rank of a \(3\times5\) matrix?
Practice Answers
Practice 1
Evaluate the standard basis vectors:
\[ T(e_1)= \begin{bmatrix} 3\\ 1 \end{bmatrix}, \qquad T(e_2)= \begin{bmatrix} 2\\ -1 \end{bmatrix}. \]Therefore,
\[ \boxed{ A= \begin{bmatrix} 3&2\\ 1&-1 \end{bmatrix}}. \]Practice 2
\[ \boxed{T(x,y)=(2x-y,\;4x+3y)}. \]Practice 3
A \(4\times3\) matrix defines
\[ \boxed{T:\mathbb{R}^3\to\mathbb{R}^4}. \]Practice 4
\[ \begin{bmatrix} 1&2\\ -1&3 \end{bmatrix} \begin{bmatrix} 4\\ -2 \end{bmatrix} = \begin{bmatrix} 0\\ -10 \end{bmatrix}. \]Thus
\[ \boxed{ \begin{bmatrix} 0\\ -10 \end{bmatrix}}. \]Practice 5
The columns are the basis images:
\[ \boxed{ T(e_1)= \begin{bmatrix} 1\\ 2\\ 0 \end{bmatrix}}, \] \[ \boxed{ T(e_2)= \begin{bmatrix} 0\\ 3\\ -1 \end{bmatrix}}. \]Practice 6
The second column is twice the first, so the image is the line
\[ \boxed{ \operatorname{span} \left\{ \begin{bmatrix} 1\\ 2 \end{bmatrix} \right\}}. \]Practice 7
Since \(T\) acts first and \(S\) second,
\[ \boxed{S\circ T\text{ is represented by }BA}. \]Practice 8
\[ \boxed{ [D]_{C\leftarrow B} = \begin{bmatrix} 0&1&0\\ 0&0&2 \end{bmatrix}}. \]Practice 9
\[ \boxed{ I_3= \begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}}. \]Practice 10
No. The second column is twice the first, so the columns are linearly dependent. The transformation cannot be one-to-one or onto \(\mathbb{R}^2\).
Practice 11
\[ x= P_B[x]_B = \begin{bmatrix} 1&1\\ 1&-1 \end{bmatrix} \begin{bmatrix} 3\\ 2 \end{bmatrix} = \boxed{ \begin{bmatrix} 5\\ 1 \end{bmatrix}}. \]Practice 12
Rank cannot exceed the smaller of the row and column counts:
\[ \boxed{\operatorname{rank}(A)\leq3}. \]Therefore the largest possible rank is \(3\).
What to Remember
A matrix represents a linear transformation relative to chosen bases.
- The standard matrix columns are \(T(e_1),\ldots,T(e_n)\).
- An \(m\times n\) matrix defines a map \(\mathbb{R}^n\to\mathbb{R}^m\).
- Matrix-vector multiplication forms a linear combination of the columns.
- The column space is the image of the transformation.
- The null space is the kernel.
- Composition corresponds to matrix multiplication.
- Inverse transformations correspond to inverse matrices.
- Different bases can produce different matrices for the same transformation.
- Basis changes are themselves implemented by invertible matrices.
The key computational identity is:
\[ \boxed{T(x)=Ax}. \]Where We Go Next
Matrix representations let us encode linear relationships compactly. The next step is learning how to solve the systems those matrices represent and how row operations expose the structure of their solution sets.
The next lesson, Systems and Elimination, develops augmented matrices, elimination, pivots, consistency, free variables, and the systematic solution of multidimensional linear systems.