Exam Questions - Matrix transformations

Matrix Transformations

A matrix is a table of numbers arranged in rows and columns.
The order of a matrix refers to its dimensions, given as rows x columns. A matrix with n rows and m columns is referred to as n x m.
A matrix can be used to represent a linear transformation of the form y = Ax, where A is the matrix, x is an input vector, and y is an output vector.
Identity matrix, denoted as I or I_n, is a special matrix with ones on the diagonal and zeros everywhere else. It serves as the multiplicative identity of matrices.

Every matrix corresponds to a certain transformation of the space, which could include rotation, reflection, dilation, and shearing, or a combination of these transformations.
A 2x2 matrix can be applied to 2-dimensional vectors, producing transformations in 2-dimensional space. Similarly, a 3x3 matrix applies to 3-dimensional vectors, producing transformations in 3-dimensions.
The effect of a matrix transformation can be visualised by applying the transformation to the unit square or unit cube and noting how its shape and position change.

A matrix transformation is applied to a vector by means of matrix multiplication. The transformed vector is obtained by multiplying the transformation matrix by the original vector.
Matrix transformations are linear operations. They follow the properties of associativity, distributivity over vector addition, and scalar multiplication.
To apply a series of transformations represented by different matrices, it suffices to multiply the matrices together to obtain a composite matrix, and then multiply this composite matrix by the original vector.

The inverse of a matrix, if it exists, undoes the transformation that the matrix represents.
The inverse of a matrix A is denoted as A⁻¹.
For a matrix to have an inverse, it must be square (same number of rows and columns) and non-singular (determinant not equal to zero).
The transformation represented by A⁻¹ is the reverse of the transformation represented by A. If A is applied first and then A⁻¹ is applied, the final position is the same as the initial position.

The determinant of a matrix is a particular number that can be calculated from its elements, and has important interpretations in terms of transformations.
For a 2x2 matrix, the determinant gives the factor by which area is scaled under the transformation. For a 3x3 matrix, the determinant gives the factor by which volume is scaled.
The sign of the determinant indicates whether the transformation maintains or reverses the orientation of space. A positive determinant means the transformation maintains orientation, while a negative determinant means it reverses orientation.

These core principles should present a good understanding of matrix transformations when preparing for further mathematical examinations.