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The image of a linear transformation is a fundamental concept in linear algebra and provides insight into the behavior and the nature of the transformation. In simple terms, the image of a linear transformation represented by a matrix is the set of all vectors that can be obtained by applying the transformation to all possible input vectors.

For the transformation given by the matrix \(A\), which is a 3 by 3 matrix, the image will tell us about the possible outputs in three-dimensional space, \(\mathbb{R}^{3}\). The image could either be all of \(\mathbb{R}^{3}\), a plane within \(\mathbb{R}^{3}\), or a line. The geometric nature of the image is intimately connected to the concept of matrix rank, which we will explore in detail.

Matrix rank is a measure of the 'non-degeneracy' of a system of linear equations corresponding to the matrix. Essentially, the rank tells us the maximum number of independent directions in the image of the linear transformation. For our matrix \(A\), rank is the key to determine the dimensionality of its image in \(\mathbb{R}^{3}\). Geometrically, if the rank of \(A\) is 3, it signifies that the image is three-dimensional, namely all of \(\mathbb{R}^{3}\).

If it has a rank of 2, the image is a two-dimensional plane. A rank of 1 indicates the image is a one-dimensional line. The rank gives us essential information, not only about the size and shape of the image but also about the system's ability to span space and, subsequently, about the solutions of the system of linear equations tied to \(A\).

Row echelon form is a particular arrangement of a matrix that provides a clearer picture of its rank. A matrix is in row echelon form when it satisfies the following conditions: all non-zero rows are above any rows of all zeroes; each leading entry of a row is in a column to the right of the leading entry of the row above it; and the leading entry in any non-zero row is 1 (called a pivot).

These conditions help identify the linearly independent rows, which correspond to the rank of the matrix. By transforming matrix \(A\) into row echelon form through a series of row operations, we expose the structure of the matrix which ultimately helps us understand the geometry of its image. For instance, if after transformations all three rows have leading entries we can conclude that the matrix has full rank and its image is all of \(\mathbb{R}^{3}\). If there are only two rows with leading entries, the third dimension is dependent on the first two, forming a plane.