The Central Idea
A function between vector spaces is linear when it respects linear combinations.
If \(T:V\to W\), then linearity means
\[ \boxed{T(c_1v_1+c_2v_2)=c_1T(v_1)+c_2T(v_2)} \]for every \(v_1,v_2\in V\) and every scalar \(c_1,c_2\).
This single principle says that the transformation preserves the linear structure of the space.
Definition of a Linear Transformation
A function
\[ T:V\to W \]between vector spaces is a linear transformation if, for all \(u,v\in V\) and scalars \(c\),
\[ \boxed{T(u+v)=T(u)+T(v)} \]and
\[ \boxed{T(cv)=cT(v)}. \]These are the additivity and homogeneity conditions.
Linearity Preserves Every Finite Linear Combination
Once additivity and scalar compatibility hold, the transformation automatically satisfies
\[ T(c_1v_1+\cdots+c_kv_k) = c_1T(v_1)+\cdots+c_kT(v_k). \]This is why bases become so powerful: once we know what a linear transformation does to basis vectors, we know what it does to every vector in the space.
Domain, Codomain, and Output
In
\[ T:V\to W, \]\(V\) is the domain and \(W\) is the codomain.
A vector \(v\in V\) is sent to an output
\[ T(v)\in W. \]The domain and codomain do not need to be the same vector space.
Example: Scaling in \(\mathbb{R}^2\)
Define
\[ T(x,y)=(2x,2y). \]Every vector is doubled in length while its direction is preserved.
For example,
\[ T(3,-1)=(6,-2). \]This transformation is linear because scaling distributes over vector addition and scalar multiplication.
Example: Unequal Coordinate Scaling
Define
\[ T(x,y)=(3x,\tfrac12 y). \]The horizontal coordinate is stretched by \(3\), while the vertical coordinate is compressed by a factor of \(1/2\).
This is still linear even though different directions are treated differently.
Example: Reflection
Reflection across the \(x\)-axis is given by
\[ T(x,y)=(x,-y). \]Reflection across the \(y\)-axis is
\[ T(x,y)=(-x,y). \]Both are linear transformations.
Example: Rotation
Rotation of the plane through an angle \(\theta\) about the origin is linear.
In coordinates,
\[ T(x,y)= \left( x\cos\theta-y\sin\theta,\; x\sin\theta+y\cos\theta \right). \]The origin remains fixed, and vector addition and scalar multiplication are preserved.
Example: Projection
Projection of \(\mathbb{R}^3\) onto the \(xy\)-plane is
\[ T(x,y,z)=(x,y,0). \]The \(z\)-component is removed.
This transformation is linear, but it is not invertible because different vectors can produce the same projection.
Example: A Coordinate Shear
Define
\[ T(x,y)=(x+y,y). \]The second coordinate is preserved, while the first coordinate is shifted by an amount proportional to \(y\).
This transformation is called a shear.
Example: From \(\mathbb{R}^2\) to \(\mathbb{R}^3\)
Define
\[ T(x,y)=(x,y,x+y). \]This map takes a two-dimensional input and produces a three-coordinate output.
Its outputs lie in a two-dimensional subspace of \(\mathbb{R}^3\), namely the plane
\[ z=x+y. \]Example: From Polynomials to Numbers
Let \(P_2\) be the space of polynomials of degree at most \(2\). Define
\[ T(p)=p(1). \]If
\[ p(x)=a+bx+cx^2, \]then
\[ T(p)=a+b+c. \]Evaluation at a fixed point is linear.
Example: Differentiation
Differentiation is a linear transformation.
For polynomials,
\[ D:P_n\to P_{n-1} \]defined by
\[ D(p)=p' \]satisfies
\[ D(p+q)=D(p)+D(q) \]and
\[ D(cp)=cD(p). \]Example: Integration with a Fixed Lower Limit
Define
\[ T(f)(x)=\int_0^x f(t)\,dt. \]Then
\[ T(af+bg)=aT(f)+bT(g). \]Thus integration with a fixed lower limit is linear on suitable function spaces.
The Zero Vector Must Map to Zero
Every linear transformation satisfies
\[ \boxed{T(0)=0}. \]To see why,
\[ T(0)=T(0+0)=T(0)+T(0). \]Subtracting \(T(0)\) from both sides gives
\[ T(0)=0. \]A Quick Nonlinearity Test
If a proposed transformation does not send the zero vector to the zero vector, it cannot be linear.
For example,
\[ T(x,y)=(x+1,y) \]satisfies
\[ T(0,0)=(1,0)\neq(0,0). \]Therefore it is not linear.
Translations Are Not Linear
A translation such as
\[ T(x,y)=(x+3,y-2) \]moves every point by a fixed amount.
It does not preserve the origin, so it is not linear.
Translations are affine transformations rather than linear transformations.
Nonlinear Example: Squaring Coordinates
Consider
\[ T(x,y)=(x^2,y^2). \]In general,
\[ T(2x,2y)\neq2T(x,y). \]The scalar-multiplication property fails, so the map is nonlinear.
Nonlinear Example: Multiplying Coordinates
The map
\[ T(x,y)=xy \]is not linear because the variables multiply one another.
Linear transformations may mix coordinates through sums and scalar multiples, but not through products of the coordinates themselves.
A Reliable Linearity Test
To test whether \(T\) is linear, check:
\[ T(u+v)=T(u)+T(v) \]and
\[ T(cu)=cT(u). \]Equivalently, test the combined condition
\[ \boxed{T(au+bv)=aT(u)+bT(v)}. \]Example: Verify Linearity
Let
\[ T(x,y)=(2x-y,3y). \]For
\[ u=(x_1,y_1),\qquad v=(x_2,y_2), \]and scalars \(a,b\),
\[ T(au+bv) \]becomes
\[ \left( 2(ax_1+bx_2)-(ay_1+by_2), 3(ay_1+by_2) \right). \]Rearranging gives
\[ aT(u)+bT(v). \]Therefore \(T\) is linear.
Linear Transformations Are Determined by a Basis
Suppose
\[ B=\{v_1,\ldots,v_n\} \]is a basis for \(V\).
Every vector \(v\in V\) has a unique representation
\[ v=c_1v_1+\cdots+c_nv_n. \]Therefore,
\[ \boxed{ T(v)=c_1T(v_1)+\cdots+c_nT(v_n)}. \]Knowing the images of the basis vectors determines the entire transformation.
Example: Define a Transformation from Basis Images
Suppose
\[ T(e_1)= \begin{bmatrix} 2\\ 1 \end{bmatrix}, \qquad T(e_2)= \begin{bmatrix} -1\\ 3 \end{bmatrix}. \]Since
\[ \begin{bmatrix} x\\ y \end{bmatrix} = xe_1+ye_2, \]linearity gives
\[ T(x,y) = x \begin{bmatrix} 2\\ 1 \end{bmatrix} + y \begin{bmatrix} -1\\ 3 \end{bmatrix}. \]Thus
\[ \boxed{T(x,y)=(2x-y,\;x+3y)}. \]Image of a Linear Transformation
The image, also called the range, of \(T:V\to W\) is the set of all outputs:
\[ \boxed{ \operatorname{Im}(T) = \{T(v):v\in V\}}. \]The image is always a subspace of the codomain \(W\).
Why the Image Is a Subspace
Suppose
\[ y_1=T(v_1),\qquad y_2=T(v_2). \]For scalars \(a,b\),
\[ ay_1+by_2 = aT(v_1)+bT(v_2) = T(av_1+bv_2). \]Therefore the linear combination remains in the image.
Kernel of a Linear Transformation
The kernel of \(T:V\to W\) is the set of inputs mapped to the zero vector:
\[ \boxed{ \ker(T) = \{v\in V:T(v)=0\}}. \]The kernel measures which input directions disappear under the transformation.
The Kernel Is a Subspace
If
\[ T(u)=0 \qquad\text{and}\qquad T(v)=0, \]then
\[ T(au+bv) = aT(u)+bT(v) = 0. \]Hence
\[ \boxed{\ker(T)\text{ is a subspace of the domain}}. \]Example: Kernel of a Projection
Let
\[ T(x,y,z)=(x,y,0). \]To find the kernel, solve
\[ T(x,y,z)=(0,0,0). \]This requires
\[ x=0,\qquad y=0, \]while \(z\) is free.
Therefore,
\[ \boxed{ \ker(T) = \operatorname{span} \left\{ \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix} \right\}}. \]Example: Image of a Projection
For the same transformation
\[ T(x,y,z)=(x,y,0), \]every output has the form
\[ \begin{bmatrix} x\\ y\\ 0 \end{bmatrix}. \]Thus
\[ \boxed{ \operatorname{Im}(T) = \operatorname{span} \left\{ \begin{bmatrix}1\\0\\0\end{bmatrix}, \begin{bmatrix}0\\1\\0\end{bmatrix} \right\}}. \]Kernel and Information Loss
A nonzero vector in the kernel is completely erased by the transformation.
If
\[ v\neq0 \qquad\text{but}\qquad T(v)=0, \]then information about movement in the direction \(v\) is lost.
This is why a nontrivial kernel prevents a transformation from being one-to-one.
One-to-One Linear Transformations
A linear transformation is one-to-one if different inputs always produce different outputs.
For a linear transformation,
\[ \boxed{ T\text{ is one-to-one} \iff \ker(T)=\{0\}}. \]The zero vector must be the only input that disappears.
Onto Linear Transformations
A transformation
\[ T:V\to W \]is onto if every vector in the codomain \(W\) is produced by at least one input.
Equivalently,
\[ \boxed{\operatorname{Im}(T)=W}. \]Invertible Linear Transformations
A linear transformation is invertible when every output corresponds to exactly one input.
This requires both:
- one-to-one behavior;
- onto behavior.
Then there exists an inverse transformation
\[ T^{-1}:W\to V. \]The Inverse of a Linear Transformation Is Linear
If \(T\) is an invertible linear transformation, then its inverse is also linear.
The transformation and its inverse preserve the same underlying linear structure in opposite directions.
Example: An Invertible Scaling
Consider
\[ T(x,y)=(2x,3y). \]No nonzero direction is collapsed, and every point in \(\mathbb{R}^2\) can be reached.
The inverse is
\[ \boxed{ T^{-1}(u,v)= \left( \frac{u}{2}, \frac{v}{3} \right)}. \]Example: A Noninvertible Projection
The projection
\[ T(x,y,z)=(x,y,0) \]is not one-to-one because
\[ T(x,y,z_1)=T(x,y,z_2) \]for any \(z_1,z_2\).
Once the \(z\)-coordinate is discarded, it cannot be recovered.
Composition of Linear Transformations
Suppose
\[ T:U\to V \]and
\[ S:V\to W \]are linear.
Their composition
\[ S\circ T:U\to W \]is defined by
\[ (S\circ T)(u)=S(T(u)). \]The composition is also linear.
Order Matters in Composition
In general,
\[ S\circ T\neq T\circ S. \]Rotating and then projecting may produce a different result from projecting and then rotating.
Linear transformations therefore compose in an ordered sequence.
The Identity Transformation
The identity transformation on \(V\) is
\[ I_V(v)=v. \]It leaves every vector unchanged.
For any linear transformation \(T:V\to W\),
\[ T\circ I_V=T \]and
\[ I_W\circ T=T. \]The Zero Transformation
The zero transformation sends every vector to the zero vector:
\[ T(v)=0. \]It is linear.
Its kernel is the entire domain, and its image contains only the zero vector.
Linear Transformations Preserve Span Relationships
If
\[ v\in\operatorname{span}\{v_1,\ldots,v_k\}, \]then
\[ T(v)\in \operatorname{span}\{T(v_1),\ldots,T(v_k)\}. \]Linear transformations carry linear combinations into corresponding linear combinations of their images.
What Happens to Independent Sets?
A linear transformation does not always preserve linear independence.
A projection can take two independent vectors and map them onto the same line.
However, an injective linear transformation does preserve independence.
Injective Maps Preserve Independence
Suppose \(T\) is one-to-one and
\[ v_1,\ldots,v_k \]are independent.
If
\[ c_1T(v_1)+\cdots+c_kT(v_k)=0, \]then
\[ T(c_1v_1+\cdots+c_kv_k)=0. \]Since \(\ker(T)=\{0\}\),
\[ c_1v_1+\cdots+c_kv_k=0. \]Independence then forces every coefficient to be zero.
Linear Transformations and Dimension
A linear transformation can preserve dimension, reduce dimension, or move a space into another space of equal or larger ambient dimension.
It cannot create more independent output directions than are present in the input space.
Rank
The dimension of the image of a linear transformation is called its rank:
\[ \boxed{\operatorname{rank}(T)=\dim(\operatorname{Im}(T))}. \]Rank measures how many independent output directions the transformation produces.
Nullity
The dimension of the kernel is called the nullity:
\[ \boxed{\operatorname{nullity}(T)=\dim(\ker(T))}. \]Nullity measures how many independent input directions are collapsed to zero.
The Rank-Nullity Relationship
For a linear transformation
\[ T:V\to W \]with finite-dimensional domain \(V\),
\[ \boxed{ \dim(V) = \operatorname{rank}(T) + \operatorname{nullity}(T)}. \]Every independent input direction either contributes to the image or is lost in the kernel.
Example: Rank and Nullity of a Projection
For
\[ T(x,y,z)=(x,y,0), \]the image is the \(xy\)-plane, so
\[ \operatorname{rank}(T)=2. \]The kernel is the \(z\)-axis, so
\[ \operatorname{nullity}(T)=1. \]Therefore,
\[ 3=2+1, \]exactly as rank-nullity predicts.
Transformations Between Spaces of Different Dimensions
If
\[ T:\mathbb{R}^2\to\mathbb{R}^3, \]the image can have dimension at most \(2\).
If
\[ T:\mathbb{R}^4\to\mathbb{R}^2, \]the image can have dimension at most \(2\), so at least two independent input directions must be lost if the rank reaches \(2\).
Linear Functionals
A linear transformation from a vector space into its scalar field is called a linear functional.
For example,
\[ T(x,y,z)=2x-y+4z \]maps \(\mathbb{R}^3\) into \(\mathbb{R}\).
Such maps appear throughout optimization, geometry, mechanics, economics, and data analysis.
Transformation of a Basis
Suppose \(B=\{v_1,\ldots,v_n\}\) is a basis for \(V\).
Then the vectors
\[ T(v_1),\ldots,T(v_n) \]determine the image of the entire space.
In fact,
\[ \boxed{ \operatorname{Im}(T) = \operatorname{span} \{T(v_1),\ldots,T(v_n)\}}. \]Why Bases Make Transformations Computable
If a vector is written as
\[ v=c_1v_1+\cdots+c_nv_n, \]then
\[ T(v) = c_1T(v_1)+\cdots+c_nT(v_n). \]Instead of defining \(T\) separately for infinitely many vectors, we only need to specify its action on a basis.
From Basis Images to Matrices
When coordinate bases are chosen, the images of the basis vectors can be placed into the columns of a matrix.
That matrix becomes a compact numerical representation of the transformation.
The next lesson develops this connection systematically.
Common Error: Assuming Every Formula Is Linear
A formula involving vectors is not automatically a linear transformation.
Terms such as
\[ x^2,\qquad xy,\qquad \sin x,\qquad |x| \]usually signal nonlinear behavior when they are applied to the input coordinates themselves.
Common Error: Forgetting the Origin Test
If
\[ T(0)\neq0, \]the transformation is immediately known to be nonlinear.
This is often the fastest first test.
Common Error: Confusing Image with Codomain
The codomain is the declared destination space.
The image is the portion of that space actually reached.
For example, a map
\[ T:\mathbb{R}^2\to\mathbb{R}^3 \]may have a two-dimensional plane as its image even though the codomain is all of \(\mathbb{R}^3\).
Common Error: Confusing Kernel with the Zero Vector
The kernel is a set of input vectors, not merely the zero output.
It may contain only \(0\), or it may contain an entire line, plane, or higher-dimensional subspace.
Common Error: Assuming Linear Means One-to-One
A linear transformation may collapse many vectors onto the same output.
Projection is linear but not one-to-one.
Linearity and invertibility are separate properties.
Common Error: Assuming Linear Means Onto
A linear map can land in only part of its codomain.
For example,
\[ T(x,y)=(x,y,x+y) \]maps into \(\mathbb{R}^3\), but its image is only a plane.
Common Error: Reversing Composition Order
The expression
\[ S\circ T \]means apply \(T\) first and then apply \(S\).
Composition is read from right to left.
A Reliable Procedure for Testing Linearity
- Identify the domain and codomain.
- Check \(T(0)=0\) as a fast necessary test.
- Take arbitrary vectors \(u,v\) and scalars \(a,b\).
- Compute \(T(au+bv)\).
- Compute \(aT(u)+bT(v)\).
- Compare the results.
- If they agree for all choices, the map is linear.
A Reliable Procedure for Finding a Kernel
- Set the transformation equal to the zero output.
- Solve \(T(v)=0\).
- Write the solution set parametrically.
- Express it as a span when possible.
- Identify a basis and dimension if needed.
A Reliable Procedure for Finding an Image
- Choose a basis for the domain.
- Compute the image of each basis vector.
- Take the span of those output vectors.
- Remove redundant output vectors if necessary.
- The remaining vectors form a basis for the image.
Example: Full Kernel and Image Workflow
Consider
\[ T:\mathbb{R}^3\to\mathbb{R}^2 \]defined by
\[ T(x,y,z)=(x+y,\;y+z). \]To find the kernel, solve
\[ x+y=0, \qquad y+z=0. \]Let \(y=t\). Then
\[ x=-t,\qquad z=-t. \]Thus
\[ \ker(T) = \left\{ t \begin{bmatrix} -1\\ 1\\ -1 \end{bmatrix} :t\in\mathbb{R} \right\}. \]Therefore,
\[ \boxed{ \ker(T) = \operatorname{span} \left\{ \begin{bmatrix} -1\\ 1\\ -1 \end{bmatrix} \right\}}. \]Next compute the images of the standard basis vectors:
\[ T(e_1)= \begin{bmatrix} 1\\ 0 \end{bmatrix}, \qquad T(e_2)= \begin{bmatrix} 1\\ 1 \end{bmatrix}, \qquad T(e_3)= \begin{bmatrix} 0\\ 1 \end{bmatrix}. \]The first and third already span \(\mathbb{R}^2\). Hence
\[ \boxed{\operatorname{Im}(T)=\mathbb{R}^2}. \]So
\[ \operatorname{rank}(T)=2, \qquad \operatorname{nullity}(T)=1, \]and
\[ 3=2+1. \]Why Linear Transformations Matter in Multidimensional Systems
Linear transformations provide the operational layer of linear algebra. Vector spaces describe allowable states; bases describe coordinates; transformations describe how those states change.
In applications, linear transformations can model sensor mappings, coordinate changes, projections, rotations, state transitions, feature extraction, signal processing, dimensional reduction, and local approximations to nonlinear systems.
Once a transformation is represented by a matrix, these abstract ideas become directly computable.
Practice 1: Identify Linearity
Is
\[ T(x,y)=(3x,-2y) \]linear?
Practice 2: Origin Test
Is
\[ T(x,y)=(x+4,y) \]linear?
Practice 3: Compute an Output
Let
\[ T(x,y)=(2x-y,x+3y). \]Find \(T(4,-1)\).
Practice 4: Basis Images
Suppose
\[ T(e_1)= \begin{bmatrix} 1\\ 2 \end{bmatrix}, \qquad T(e_2)= \begin{bmatrix} 3\\ -1 \end{bmatrix}. \]Find
\[ T \left( \begin{bmatrix} 2\\ 5 \end{bmatrix} \right). \]Practice 5: Kernel
Find the kernel of
\[ T(x,y)=(x+y,0). \]Practice 6: Image
Describe the image of
\[ T(x,y,z)=(x,y,0). \]Practice 7: One-to-One?
Is the projection
\[ T(x,y,z)=(x,y,0) \]one-to-one?
Practice 8: Onto?
Is
\[ T:\mathbb{R}^2\to\mathbb{R}^3, \qquad T(x,y)=(x,y,x+y) \]onto \(\mathbb{R}^3\)?
Practice 9: Differentiation
Let
\[ D:P_3\to P_2 \]be differentiation. Find
\[ D(2-3x+5x^2-x^3). \]Practice 10: Composition
Let
\[ T(x,y)=(2x,y) \]and
\[ S(x,y)=(x,-y). \]Find \((S\circ T)(x,y)\).
Practice 11: Rank and Nullity
Suppose
\[ T:\mathbb{R}^4\to\mathbb{R}^3 \]has rank \(3\). What is its nullity?
Practice 12: Linear or Nonlinear?
Determine whether
\[ T(x,y)=(x,xy) \]is linear.
Practice Answers
Practice 1
Yes. Each coordinate is a scalar multiple of an input coordinate, and the transformation preserves linear combinations.
Practice 2
No. Since
\[ T(0,0)=(4,0)\neq(0,0), \]the transformation is not linear.
Practice 3
\[ T(4,-1) = (2(4)-(-1),\;4+3(-1)) = \boxed{(9,1)}. \]Practice 4
Since
\[ \begin{bmatrix} 2\\ 5 \end{bmatrix} = 2e_1+5e_2, \]linearity gives
\[ T \left( \begin{bmatrix} 2\\ 5 \end{bmatrix} \right) = 2 \begin{bmatrix} 1\\ 2 \end{bmatrix} + 5 \begin{bmatrix} 3\\ -1 \end{bmatrix} = \boxed{ \begin{bmatrix} 17\\ -1 \end{bmatrix}}. \]Practice 5
Solve
\[ x+y=0. \]Thus \(x=-y\). Let \(y=t\):
\[ \boxed{ \ker(T) = \operatorname{span} \left\{ \begin{bmatrix} -1\\ 1 \end{bmatrix} \right\}}. \]Practice 6
Every output has the form
\[ (x,y,0). \]Therefore the image is the \(xy\)-plane:
\[ \boxed{ \operatorname{Im}(T) = \operatorname{span} \left\{ \begin{bmatrix}1\\0\\0\end{bmatrix}, \begin{bmatrix}0\\1\\0\end{bmatrix} \right\}}. \]Practice 7
No. Different vectors with different \(z\)-coordinates can produce the same output.
Practice 8
No. The image is the plane
\[ z=x+y, \]not all of \(\mathbb{R}^3\).
Practice 9
\[ \boxed{ D(2-3x+5x^2-x^3) = -3+10x-3x^2}. \]Practice 10
Apply \(T\) first:
\[ T(x,y)=(2x,y). \]Then apply \(S\):
\[ \boxed{(S\circ T)(x,y)=(2x,-y)}. \]Practice 11
By rank-nullity,
\[ 4=3+\operatorname{nullity}(T). \]Therefore,
\[ \boxed{\operatorname{nullity}(T)=1}. \]Practice 12
No. The product \(xy\) introduces nonlinear behavior. For example, homogeneity fails in general.
What to Remember
A linear transformation preserves vector addition and scalar multiplication:
\[ \boxed{T(au+bv)=aT(u)+bT(v)}. \]- Every linear transformation sends \(0\) to \(0\).
- The image is the subspace of outputs actually reached.
- The kernel is the subspace of inputs mapped to zero.
- A linear transformation is one-to-one exactly when its kernel is \(\{0\}\).
- It is onto when its image equals the codomain.
- An invertible linear transformation is both one-to-one and onto.
- Rank is the dimension of the image.
- Nullity is the dimension of the kernel.
- For finite-dimensional domains, rank plus nullity equals domain dimension.
- A linear transformation is completely determined by what it does to a basis.
The key operational idea is:
\[ \boxed{\text{linear maps preserve linear combinations}}. \]Where We Go Next
A linear transformation can be described abstractly by its action on vectors, but computation becomes much easier once coordinates are chosen.
The next lesson, Matrix Representations, shows how the images of basis vectors become matrix columns and how matrix multiplication implements linear transformations numerically.