Verify both parts of Theorem 1.16 for the scalar, vectors,...

Verify both parts of Theorem $1.16$ for the scalar, vectors, and matrix given .
$a=2, A=\left(\begin{array}{rr}-1 & 2 \\ 3 & 0\end{array}\right), \mathbf{x}=\left(\begin{array}{r}1 \\ -2\end{array}\right), \mathbf{y}=\left(\begin{array}{l}0 \\ 4\end{array}\right)$

Key Concepts

Distributive Property of Scalar Multiplication over Addition

This concept asserts that when a scalar multiplies a sum of elements (whether they be numbers, vectors, or matrices), the result is the same as multiplying the scalar with each individual element and then adding those results. It guarantees that for any scalar k and any two elements x and y in a linear structure, k(x + y) equals kx + ky, which is fundamental to the linear behavior of vector spaces and matrix operations.

Distributive Property of Scalar Addition over Multiplication

This principle states that when two scalars are added first and then multiplied by an element (like a vector or matrix), the outcome is the same as multiplying the element by each scalar separately and then adding the products. In formal terms, for any scalars a and b and any element x, (a + b)x equals ax + bx, ensuring consistency in the interplay between scalar arithmetic and linear transformations.

Linearity of Matrix Operations

Matrix operations, including both matrix addition and matrix-vector multiplication, are based on the principle of linearity. This means that matrices preserve the structure of addition and scalar multiplication as observed in their interactions with vectors and scalars. This property is crucial for understanding how matrices represent linear transformations and how they combine with vectors in solving systems and modeling relationships in linear algebra.

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