Justify the following alternative formula for multiplying a...

Justify the following alternative formula for multiplying a matrix $A$ and a column vector $\mathbf{x}$ : $$ A \mathbf{x}=x_1 \mathbf{c}_1+x_2 \mathbf{c}_2+\cdots+x_n \mathbf{c}_n, $$ where $\mathbf{c}_1, \ldots, \mathbf{c}_n$ are the columns of $A$ and $x_1, \ldots, x_n$ the entries of $\mathbf{x}$.

Justify the following alternative formula for multiplying a matrix $A$ and a column vector $\mathbf{x}$ :
$$
A \mathbf{x}=x_1 \mathbf{c}_1+x_2 \mathbf{c}_2+\cdots+x_n \mathbf{c}_n,
$$
where $\mathbf{c}_1, \ldots, \mathbf{c}_n$ are the columns of $A$ and $x_1, \ldots, x_n$ the entries of $\mathbf{x}$.

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Key Concepts

Matrix-Vector Multiplication

Matrix-vector multiplication is a fundamental operation where each entry of the resulting vector is computed as the dot product of a row of the matrix with the vector. This definition underpins the idea that a matrix represents a linear transformation acting on a vector space, and the operation combines the entries of the vector with the corresponding entries in the matrix.

Linear Combination

A linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results together. In the context of a matrix and a vector, the product A x can be viewed as a linear combination of the matrix's column vectors, where each scalar from the vector multiplies its corresponding column, highlighting how the transformation aggregates contributions from each column.

Distributive Property of Matrix Multiplication

The distributive property in matrix multiplication allows the expression of the product as the sum of the products of the matrix's columns and the corresponding entries of the vector. This property justifies rewriting the matrix-vector product as x_1 c_1 + x_2 c_2 + ... + x_n c_n, demonstrating how each entry in the vector scales its respective column in the matrix, thereby preserving the linear structure of the operation.

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