Final answer:
The expressions involving vector operations must be corrected based on vector algebra principles. Traditional multiplication as implied by AB is not defined for vectors. Instead, we use operations like dot product (A · B), cross product (A × B), addition (A + B), and subtraction (A - B) appropriate for the context of vectors.
Step-by-step explanation:
The question posed by the student involves understanding various operations that can be performed on vectors. Considering the linear transformation matrix operations, we must correct these expressions based on the principles of vector algebra.
Issues with the Given Expressions
- For (a) C = AB, unless A and B are specifically matrices representing linear transformations, this expression is generally undefined as vectors do not have a multiplication operation like matrices.
- In (b) C = A + B, this is a standard operation representing adding two vectors.
- For (c) C = A × B, this indicates the cross product between two vectors, resulting in a vector perpendicular to A and B.
- Regarding (d) C = A - B, this represents the subtraction of vector B from vector A, giving the vector that points from the tip of B to the tip of A.
To correct them according to the context of linear transformations:
- C = A · B represents the dot product if A and B are vectors, resulting in a scalar.
- C = A × B can stand as the cross product, or if subtraction is intended, C = A - B would be correct for vector subtraction.
- C = A × B is correct as the cross product of two vectors.
- C = A - B is already accurate for vector subtraction.
When dealing with vectors, the context is crucial for determining whether to perform scalar multiplication, dot product, cross product, addition, or subtraction.