Let h, k>0 be fixed. Let S={(i h, j k) |1 ≤i ≤m, 1 ≤j ≤n} be points in a rectangular planar grid

step 1 We have that each column 1 through m of A will have to be a multiple of its respective vector, v

Let h, k>0 be fixed. Let S={(i h, j k) |1 ≤i ≤m, 1 ≤j ≤n} be...

Let $h, k>0$ be fixed. Let $S=\{(i h, j k) \mid 1 \leq i \leq m, 1 \leq j \leq n\}$ be points in a rectangular planar grid. Show that the function space $\mathcal{F}(S)$ can be identified with the vector space of $m \times n$ matrices $\mathcal{M}_{m \times n}$.

Key Concepts

Basis Identification

Both F(S) and M??? have natural bases that reflect their structure. For F(S), the basis can be taken as the set of indicator functions for each point in S, while for M??? the basis consists of matrices with a single 1 in one position and zeros elsewhere. The correspondence between these bases under the isomorphism further justifies the identification of the function space with the matrix space.

Function Space

A function space is the set of all functions from a given set to a field (typically the real numbers) that is equipped with operations like pointwise addition and scalar multiplication. In this context, the function space F(S) contains every possible function defined on the discrete set S of grid points, making it a vector space where each function is completely determined by its values at the m × n points.

Vector Space Isomorphism

A vector space isomorphism is a bijective linear map between two vector spaces that preserves the operations of addition and scalar multiplication. Since both F(S) and the space of m × n matrices have the same finite dimension (m × n), they are isomorphic. This means that there exists a natural correspondence between a function's values over the grid and the entries of a matrix.

Matrix Representation

The process of matrix representation involves associating each function in F(S) with an m × n matrix, where the entry in the i?th row and j?th column corresponds to the function’s value at the point (ih, jk). This creates a concrete representation of abstract functions, using a familiar structure from linear algebra.

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