Maths (university) · Part 08
Vectors, matrices and linear systems
Arrows, tables of numbers and how to solve several equations at once.
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- VectorA quantity with a size and a direction, written as a list of numbers such as (3, 4). Its length is the square root of the sum of squares of its entries.
- Dot productFor vectors a and b, multiply matching entries and add. It is zero exactly when the vectors are at right angles, and it equals |a||b| cos(angle).
- Cross productFor two vectors in 3D, the cross product is a new vector at right angles to both. Its length is the area of the parallelogram they span.
- MatrixA rectangular table of numbers with rows and columns. A matrix with m rows and n columns is called m by n.
- Matrix multiplicationEach entry of the product is a row of the first matrix dotted with a column of the second. The first matrix's columns must equal the second's rows.
- Linear systemA set of equations where each unknown appears only to the first power, such as x + y = 5 and x - y = 1. A solution makes every equation true at once.
- Augmented matrixA matrix that stores a system's coefficients and its right-hand sides together, separated by a bar. Row operations on it do not change the solutions.
- Gaussian eliminationUse row operations to create zeros below each leading entry, until the system is in staircase (echelon) form. Then solve from the bottom row upwards.
- Number of solutionsA linear system has exactly one solution, no solution, or infinitely many. During elimination, a row 0 = 5 means none, and a row of zeros means free variables.