Common Mistakes

Frequent Errors

1. Calling a rule linear when it adds a fixed constant unrelated to inputs (4.12.A.1). 2. Forgetting L(0)=0 (4.12.A.2). 3. Using row vectors instead of column 2x1 form (4.12.A.3). 4. Building A from wrong basis images (4.12.A.4). 5. Miscomputing Av with matrix multiply errors (4.12.A.4, 4.10.A.2). 6. Wrong layout for 2xn batch inputs (4.12.A.5). 7. Assuming any function on vectors is linear. 8. Confusing L(v)=Av with element-wise operations (4.12.A.4).

Why These Happen

Students focus on numeric answers and skip the linearity definition.

How to Fix Them

• Test L(0)=0 and a simple scaling property.
• Use column matrices consistently.
• Build A from where (1,0) and (0,1) map.

Avoidance Scripts

• Zero in, zero out.
• Columns for vectors.
• Basis images build the matrix.

Quick Self-Check

• Is the rule linear?
• Did I use column form?
• Does Av match the stated rule?