Question

Using scalar notation, write out and verify the Linearity Principle

Answer 1

The Linearity Principle, expressed in scalar notation, can be written as follows:

`\[ F(c\mathbf{u} + d\mathbf{v}) = cF(\mathbf{u}) + dF(\mathbf{v}) \]`

Step-by-step explanation:

The Linearity Principle in scalar notation is a fundamental concept in linear algebra, stating that a linear transformation (F) satisfies the property of additivity and homogeneity. In the given expression,
`\( \mathbf{u} \)` and
`\( \mathbf{v} \)` are vectors, and (c) and (d) are scalars. The principle asserts that applying the linear transformation to a linear combination of vectors
`(\( c\mathbf{u} + d\mathbf{v} \))` is equivalent to the same linear combination of the transformation applied to individual vectors.

In the expression,
`\( F(c\mathbf{u} + d\mathbf{v}) \)` represents the linear transformation applied to the combined vector, while
`\( cF(\mathbf{u}) + dF(\mathbf{v}) \)` represents the linear combination of the transformation applied to each individual vector. The principle is verified by ensuring that both sides of the equation yield the same result for any choice of vectors
`\( \mathbf{u} \) and \( \mathbf{v} \)` and scalars (c) and (d).

This principle is foundational in various mathematical and scientific disciplines, providing a powerful tool for analyzing and understanding linear transformations. It underpins many concepts in linear algebra, making it a crucial element in the study of vector spaces and linear mappings.