Testing Linearity
students, in linear algebra, not every rule that turns one vector into another is a linear transformation. Some rules stretch and rotate vectors in a clean, predictable way, while others bend, shift, or distort them. 🔎 In this lesson, you will learn how to test whether a rule is linear, why that matters, and how to connect this idea to the bigger topic of linear transformations.
What Does It Mean to Be Linear?
A transformation is a rule that takes an input vector and produces an output vector. If the transformation is linear, it behaves nicely with respect to two key ideas:
- Additivity: the transformation of a sum equals the sum of the transformations.
- Homogeneity: the transformation of a scalar multiple equals the same scalar times the transformation.
In symbols, a transformation $T$ is linear if for all vectors $u$ and $v$ and all scalars $c$,
$$T(u+v)=T(u)+T(v)$$
and
$$T(cu)=cT(u).$$
These two rules capture the idea that a linear transformation preserves vector addition and scaling. Together, they imply a very important fact:
$$T(au+bv)=aT(u)+bT(v)$$
for any scalars $a$ and $b$.
This property is the reason linear transformations are so useful. They model many real systems, such as resizing an image, projecting a 3D object onto a screen, or converting coordinates in geometry. 📐
How to Test Linearity Step by Step
When you are given a rule and asked whether it is linear, students, the most reliable method is to test the two core conditions directly.
Step 1: Check the zero vector
A linear transformation must send the zero vector to the zero vector:
$$T(0)=0.$$
This is a quick first test. If a rule fails here, it is definitely not linear.
For example, suppose
$$T(x)=2x+3.$$
Then
$$T(0)=2(0)+3=3,$$
which is not $0$. So this rule is not linear.
Step 2: Check additivity
Choose two general inputs, such as $u$ and $v$, and compute both sides of
$$T(u+v)=T(u)+T(v).$$
If they are always equal, the rule passes this test.
Step 3: Check homogeneity
Choose a scalar $c$ and a vector $u$, then compare
$$T(cu)$$
with
$$cT(u).$$
If they are always equal, the rule passes this test too.
Step 4: Conclude carefully
If both properties hold for all inputs, the transformation is linear. If even one fails, it is not linear.
A common shortcut is to use the combined rule
$$T(au+bv)=aT(u)+bT(v).$$
If this fails, the transformation is not linear. If it holds for all choices, the transformation is linear.
Example 1: A Linear Transformation in $\mathbb{R}^2$
Consider the rule
$$T(x,y)=(2x-y,\,3x+4y).$$
We will test linearity.
Let $u=(x_1,y_1)$ and $v=(x_2,y_2)$. Then
$$u+v=(x_1+x_2,\,y_1+y_2).$$
Now apply $T$:
$$T(u+v)=\bigl(2(x_1+x_2)-(y_1+y_2),\,3(x_1+x_2)+4(y_1+y_2)\bigr).$$
Distribute terms:
$$T(u+v)=\bigl((2x_1-y_1)+(2x_2-y_2),\,(3x_1+4y_1)+(3x_2+4y_2)\bigr).$$
But
$$T(u)=(2x_1-y_1,\,3x_1+4y_1)$$
and
$$T(v)=(2x_2-y_2,\,3x_2+4y_2).$$
So
$$T(u)+T(v)=\bigl((2x_1-y_1)+(2x_2-y_2),\,(3x_1+4y_1)+(3x_2+4y_2)\bigr).$$
Thus
$$T(u+v)=T(u)+T(v).$$
You can also check homogeneity:
$$T(c(x,y))=T(cx,cy)=(2cx-cy,\,3cx+4cy)=c(2x-y,\,3x+4y)=cT(x,y).$$
Because both properties hold, this rule is linear. ✅
This example shows an important pattern: any transformation made only from multiplication by constants and addition of coordinates is a strong candidate for being linear.
Example 2: A Rule That Is Not Linear
Now consider
$$S(x,y)=(x+1,\,y).$$
This rule shifts every vector one unit to the right. That sounds simple, but it is not linear.
Check the zero vector:
$$S(0,0)=(1,0),$$
which is not the zero vector $(0,0)$.
So $S$ is not linear.
Why does this matter? Because linear transformations must preserve the origin. A shift changes the origin, so it breaks linearity. This is a very common way to spot nonlinearity. 🚫
Another non-linear example is
$$R(x,y)=(x^2,y).$$
Test homogeneity with scalar $c$:
$$R(c(x,y))=(c^2x^2,\,cy),$$
but
$$cR(x,y)=(cx^2,\,cy).$$
These are not equal in general, because $c^2x^2$ is usually not the same as $cx^2$. Therefore, $R$ is not linear.
Matrix Form and Why It Helps
A major fact in linear algebra is that every linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ can be written as multiplication by a matrix.
For example, if
$$T(x,y)=(2x-y,\,3x+4y),$$
then
$$T\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}2&-1\\3&4\end{pmatrix}\begin{pmatrix}x\y\end{pmatrix}.$$
This is useful because matrix multiplication automatically preserves additivity and homogeneity. That means if a rule can be written as multiplication by a matrix with no extra constant term, it is linear.
If the rule contains a constant shift like
$$T(x,y)=(2x-y+5,\,3x+4y),$$
then it is not linear, because the $+5$ cannot come from matrix multiplication alone.
So one practical strategy is:
- Look for constant terms.
- Look for powers like $x^2$ or $y^3$.
- Look for products like $xy$.
- Look for square roots, absolute values, or trig terms that are not in a linear matrix form.
These usually signal that the transformation is not linear.
Common Mistakes When Testing Linearity
students, students often make a few predictable mistakes when testing linearity. Knowing them helps you avoid errors. 🧠
Mistake 1: Testing only one example
If a rule works for one input, that does not prove it is linear. You must know it works for all inputs.
For example, a transformation might satisfy
$$T(u+v)=T(u)+T(v)$$
for one pair of vectors, but fail for another pair.
Mistake 2: Forgetting the zero vector test
If
$$T(0)\neq 0,$$
then the transformation is not linear. This is one of the fastest checks available.
Mistake 3: Confusing linear with straight-line graphs
In one-variable algebra, the word “linear” often refers to graphs like
$$y=mx+b.$$
But in linear algebra, a linear transformation must satisfy the vector rules above. A graph can be a straight line and still represent a non-linear transformation if it includes a shift.
Mistake 4: Ignoring all variables
Sometimes a rule looks complicated, but it is still linear because every output is a sum of scalar multiples of the inputs.
For example,
$$T(x,y,z)=(x-2y+3z,\,4x+z).$$
This is linear because each component is built only from additions and scalar multiplications.
Why Testing Linearity Matters
Testing linearity is not just a technical exercise. It helps you decide whether all the powerful tools of linear algebra apply.
If a transformation is linear, you can use matrices, systems of equations, bases, dimensions, and eigenvalues to study it. You can also predict how it behaves on any vector by knowing its effect on a basis.
If a transformation is not linear, those tools may not work. For example, translations and curved transformations do not behave like matrix multiplication. That means they require different methods.
In real life, linear transformations are used as approximations too. A system may not be perfectly linear, but near a certain point it may behave almost like one. This idea is important in science, engineering, and computer graphics. 📊
Conclusion
Testing linearity means checking whether a transformation preserves addition and scalar multiplication. students, if a rule satisfies
$$T(u+v)=T(u)+T(v)$$
and
$$T(cu)=cT(u),$$
then it is linear. A quick check is to see whether it sends the zero vector to the zero vector. If there are constant shifts, powers, or other nonlinear features, the rule is usually not linear.
This lesson connects directly to the broader study of linear transformations because linearity is the property that makes matrix methods work. Once you can test linearity confidently, you are ready to analyze transformations more deeply and use the full power of linear algebra. ✅
Study Notes
- A transformation is linear if it satisfies additivity and homogeneity.
- The two key equations are $T(u+v)=T(u)+T(v)$ and $T(cu)=cT(u)$.
- A linear transformation must satisfy $T(0)=0$.
- A transformation with a constant shift, such as $T(x,y)=(x+1,y)$, is not linear.
- A transformation with powers, products, or other nonlinear terms is usually not linear.
- If a rule can be written as multiplication by a matrix, it is linear.
- Linear transformations preserve structure and are central to matrix methods in linear algebra.
- Always test with general vectors and scalars, not just one example.
- Linearity connects directly to the broader topic of linear transformations because it determines when matrix tools can be used.