Final answer:
To prove a transformation is linear, one must verify closure under addition and scalar multiplication, ensure the zero vector is preserved, and show that vector addition is preserved. This involves demonstrating the commutativity of vector addition, understanding the effects of scalar multiplication on vectors, and distinguishing between vector and scalar equations.
Step-by-step explanation:
To show that a transformation is linear, you need to: a) Demonstrate closure under addition by showing that the transformation of a sum of two vectors is equal to the sum of the transformations of those vectors; b) Verify closure under scalar multiplication by proving that multiplying a vector by a scalar and then transforming it is the same as transforming the vector first and then multiplying the transformation by the scalar; c) Prove the preservation of the zero vector by confirming that transforming the zero vector yields the zero vector in the codomain; d) Establish the preservation of vector addition, which means that a transformation is linear if it preserves the addition of vectors.
For instance, if we have vectors A, B, and C, we can show that A + B + C remains unchanged regardless of the order in which the vectors are added. This property is known as the commutativity of vector addition. For example, summing in the order of A + (B + C) should yield the same result as (C + A) + B. To demonstrate this, you can choose any three vectors, add them in these different orders, and verify that the resultant vectors are equal.
Regarding the multiplication of a vector by a scalar, it affects both the magnitude and direction of the vector. In a one-dimensional context, vector quantities are added or subtracted by simply adding or subtracting their magnitudes. Geometrically in a plane, the addition or subtraction of vectors can be represented by positioning the tail of the second vector at the head of the first and then drawing a vector from the tail of the first vector to the head of the second vector. This resultant vector represents the sum or difference of the original vectors.
A vector equation represents a relationship between vector quantities, while a scalar equation represents a relationship between scalar quantities. Scalar and vector equations follow different rules due to the directional nature of vectors.