How to Verify if a Matrix Represents a Linear Transformation

What You're Actually Checking

Every matrix multiplication defines a linear transformation. But not every "formula" that produces vectors from vectors is a linear transformation. That's the distinction that matters.

Before you waste time on a problem, know this: linear transformations must satisfy two exact conditions. Everything else is just math homework with extra steps.

The Two Properties That Make the Cut

No exceptions. No workarounds. If a transformation fails either one, it's not linear.

1. Additivity

For all vectors u and v:

T(u + v) = T(u) + T(v)

Simple. Add the inputs first, get the same result as adding the outputs first.

2. Scalar Multiplication (Homogeneity)

For any vector v and scalar c:

T(cv) = cT(v)

Scale first or scale after. The result stays the same.

The Quick Filter: Zero Vector Test

Here's a fast way to catch obvious failures. Plug in the zero vector:

T(0) = 0

If your transformation doesn't map the zero vector to zero, you can stop there. It's not linear. Full stop.

This test is necessary but not sufficient. Plenty of non-linear transformations pass this test. You still need to verify both properties above.

How to Actually Verify It

Follow these steps in order. Don't skip the first one.

Step 1: Test the Zero Vector

Compute T(0). If T(0) ≠ 0, you're done. Not linear.

Step 2: Test Additivity

Pick two arbitrary vectors u and v. Compute T(u + v) and T(u) + T(v). If they're equal, additivity holds. If not, you have your answer.

Step 3: Test Scalar Multiplication

Pick an arbitrary vector v and scalar c. Compute T(cv) and cT(v). If they match, homogeneity holds.

Use specific numbers for c. The values 2, -1, and 0 work well. Zero is especially useful because it tests both properties at once.

Example Walkthrough

Let's verify whether T(x, y) = (x + 1, y) is linear.

Step 1: Zero vector test

T(0, 0) = (0 + 1, 0) = (1, 0)

Result: (1, 0), not (0, 0). Already failed. Not linear.

The "+1" constant breaks the zero vector condition immediately. No further testing needed.

Let's try one that passes Step 1. Verify T(x, y) = (2x, 3y).

Step 1: Zero vector

T(0, 0) = (0, 0). Pass.

Step 2: Additivity

Let u = (u₁, u₂), v = (v₁, v₂)

T(u + v) = T(u₁ + v₁, u₂ + v₂) = (2(u₁ + v₁), 3(u₂ + v₂)) = (2u₁ + 2v₁, 3u₂ + 3v₂)

T(u) + T(v) = (2u₁, 3u₂) + (2v₁, 3v₂) = (2u₁ + 2v₁, 3u₂ + 3v₂)

Equal. Additivity holds.

Step 3: Scalar multiplication

T(c · (x, y)) = T(cx, cy) = (2cx, 3cy) = c(2x, 3y) = cT(x, y)

Equal. Homogeneity holds.

Result: This transformation is linear.

Common Mistakes That Cost Points

• Skipping the zero vector test. It's the fastest check. Use it first.
• Using the same vector twice when testing additivity. Pick two different arbitrary vectors.
• Testing with only positive scalars. Include negative values and zero.
• Confusing linear transformations with affine transformations. Affine ones have a constant term added. That's not linear.
• Forgetting that linear transformations always include the origin. If there's a translation, it's affine, not linear.

Linear vs. Affine: The Distinction That Matters

Students mix these up constantly. Here's the difference:

• Linear: T(0) = 0. No translation. Passes through origin.
• Affine: T(0) ≠ 0. Has a translation component. T(x) = Ax + b.

If you see "+b" or any constant added to the result, it's affine. Not linear. That's not opinion—it's definition.

Quick Reference: Verification Checklist

How to Verify: Step-by-Step Process

1. Write down the transformation T(x) clearly.
2. Compute T(0). If T(0) ≠ 0, declare "not linear" and end.
3. Choose two arbitrary vectors u and v. Write them as variables.
4. Compute T(u + v) and simplify.
5. Compute T(u) + T(v) and simplify.
6. Compare results. If equal, continue. If not, declare "not linear."
7. Choose an arbitrary scalar c and vector v.
8. Compute T(cv) and simplify.
9. Compute cT(v) and simplify.
10. Compare results. If equal, it's linear. If not, it's not linear.

What About Matrix Verification?

If you're given a matrix and asked whether it represents a linear transformation, the answer is almost always yes. Every m×n matrix defines a linear transformation from ℝⁿ to ℝᵐ through matrix multiplication.

The verification work happens when you're given a formula and need to confirm it matches matrix behavior. That's when you run the tests above.

For matrix multiplication itself, the two properties are guaranteed by how matrix multiplication works. No need to re-check.

The Bottom Line

Verifying a linear transformation comes down to two properties: additivity and scalar multiplication. The zero vector test filters out obvious failures fast. Use specific vectors with actual numbers for your tests—general proof is fine, but concrete examples catch your own mistakes.

Stop overcomplicating this. The definition is the definition. Check the conditions. Done.