Revision NotesA2

matrix — A matrix is an array of numbers, usually written inside curved brackets.

Matrices are used to represent data or transformations. They consist of rows and columns, with individual entries called elements. Think of a matrix like a spreadsheet or a grid of numbers, where each cell holds a specific value and its position matters.

elements — The entries in the various cells of a matrix are known as elements.

Each number or variable within a matrix is an element. Their position within the matrix (row and column index) is important for matrix operations and defining the matrix's order. If a matrix is a bookshelf, then each book on the shelf is an element.

order — The order of a matrix is stated by the number of rows first, then the number of columns.

For example, a 5 × 5 matrix has five rows and five columns. The order is fundamental as it determines whether matrices can be added, subtracted, or multiplied. The order of a matrix is like the dimensions of a room – length by width.

square matrices — Matrices that have the same number of rows as columns are called square matrices.

Examples include 2 × 2 or 3 × 3 matrices. Square matrices are important for operations like finding determinants, inverses, and representing certain transformations. A square matrix is like a square grid or a chessboard, where the number of rows equals the number of columns.

identity matrix — An identity matrix is a square matrix with 1s on the leading diagonal and zeros elsewhere, usually denoted by I.

When an identity matrix multiplies another matrix, the other matrix remains unchanged, similar to how multiplying a number by 1 leaves it unchanged. Identity matrices must be square. The identity matrix is like the number '1' in scalar multiplication; it leaves the other matrix unchanged when multiplied.

unit matrix — The unit matrix is another name for the identity matrix.

It serves the same purpose as the identity matrix, leaving other matrices unchanged upon multiplication. It is always a square matrix with ones on the main diagonal and zeros elsewhere. Just like a 'unit' of measurement is a standard, the unit matrix is the standard 'multiplicative identity' for matrices.

zero matrix — A zero matrix is a matrix where all elements are zero, usually denoted by O.

When a zero matrix is added to another matrix, the other matrix remains unchanged, similar to how adding 0 to a number leaves it unchanged. Zero matrices can be of any order. The zero matrix is like the number '0' in scalar addition; it leaves the other matrix unchanged when added.

equal matrices — Two matrices are said to be equal if, and only if, they have the same order and each element in one matrix is equal to the corresponding element in the other matrix.

This means that for matrices A and B to be equal, they must have the same number of rows and columns, and aᵢⱼ must equal bᵢⱼ for all i and j. Two matrices being equal is like having two identical jigsaw puzzles; they must have the same number of pieces (order) and each piece must match its corresponding piece in the other puzzle (elements).

non-conformable for addition — Matrices are non-conformable for addition if they are not of the same order.

Matrix addition and subtraction require matrices to have identical dimensions (same number of rows and columns) because elements are added or subtracted in corresponding positions. Trying to add non-conformable matrices is like trying to add apples and oranges; they are different types of items and cannot be combined directly.

scalar number — A scalar number is a single number by which a matrix can be multiplied.

When a matrix is multiplied by a scalar, every element within the matrix is multiplied by that scalar. This operation changes the magnitude of the matrix elements but not its order. Multiplying a matrix by a scalar is like scaling a recipe; if you double the recipe (scalar factor of 2), you double every ingredient (element).

conformable for multiplication — Two matrices are conformable for multiplication if the number of columns in the first matrix equals the number of rows in the second matrix.

For matrices A (order m × n) and B (order n × p), the product AB exists because the 'middle' numbers (n) are the same. The resulting product matrix will have the order m × p. Being conformable for multiplication is like connecting two gears; the number of teeth on the output of the first gear must match the number of teeth on the input of the second gear for them to mesh.

object — The original point, or shape, before a transformation is called the object.

In geometric transformations, the object is the starting figure. Its coordinates or shape are altered by the transformation to produce the image. The object is like the original photograph before you apply any filters or edits.

image — The new point, or shape, after the transformation, is called the image.

The image is the result of applying a transformation to an object. Its position, orientation, or size may differ from the original object. The image is like the edited photograph after you've applied filters or made changes.

transformation — A transformation is a mapping of an object onto its image.

Transformations describe how points or shapes move in a coordinate system. They can be represented by matrices and include reflections, rotations, enlargements, stretches, and shears. A transformation is like a set of instructions for moving or changing a piece on a chessboard; it tells you exactly where the piece ends up.

unit vectors — Vectors that have length or magnitude of 1 are called unit vectors.

In two dimensions, i = (1, 0) and j = (0, 1) are unit vectors along the x-axis and y-axis, respectively. Their images under a transformation matrix form the columns of that matrix. Unit vectors are like the basic building blocks or fundamental directions (north, east) in a coordinate system, each having a standard length of one step.

linear transformations — In a linear transformation, straight lines are mapped to straight lines, and the origin is mapped to itself.

All transformations represented by matrices (reflections, rotations, enlargements, stretches, shears) are examples of linear transformations. This property ensures that the 'straightness' of objects is preserved. A linear transformation is like drawing on a rubber sheet that you can stretch or twist, but you can't tear it or fold it over itself; straight lines remain straight.

shear — A shear is a transformation where points on a fixed line stay the same, and other points move parallel to the fixed line by a distance proportional to their perpendicular distance from the fixed line.

For example, a shear with the x-axis fixed transforms the unit vector j=(0,1) to (k,1), where k is the shear factor. It distorts shapes, turning rectangles into parallelograms. Imagine pushing the top of a stack of cards while keeping the bottom card fixed; the cards slide past each other, creating a slanted stack. This is similar to a shear.

invariant points — Points that map to themselves under a transformation are called invariant points.

For a point (x, y) to be invariant under a transformation represented by matrix M, M(x, y) must equal (x, y). The origin (0,0) is always an invariant point for transformations represented by matrices. An invariant point is like a pivot in a rotation; it's the only point that doesn't move.

invariant line — A line AB is known as an invariant line under a transformation if the image of every point on AB is also on AB.

It is not necessary for each point on the line to map to itself; it can map to another point on the same line. For example, in a reflection, the mirror line is an invariant line of invariant points, while lines perpendicular to the mirror line are invariant lines but not lines of invariant points. An invariant line is like a track on which a train moves; the train (point) moves, but it stays on the track (line).