Linear Algebra: Matrices

We begin with vectors with two components (x, y), representing a point in a plane. The algebra of 2-dimensional vectors and 2×2 matrices is developed with considerable emphasis on geometric interpretation.

Having mastered 2×2 matrices we move on to 3 dimensions with 3×3 matrices and 3×3 determinants. We emphasise the geometry behind determinants by defining 2×2 determinants as areas and 3×3 determinants as volumes.

Then we move on to the general n×n case. Most matrix theory generalises readily to the n×n case but we need to define n×n determinants in a non-geometric way since we have no intuitive concept of higher dimensional spaces.

When n is bigger than 3 we have little use for n×n matrices as a geometric tool. However one of their big roles lies in the analysis of systems of linear equations. Algorithms for solving such systems are presented and are analysed using matrices.

An important concept of matrix algebra is that of eigenvalues and eigenvectors, numbers and vectors that can be associated with a square matrix that prove to be extremely useful. They can be used in a process called diagonalisation, where most n×n matrices can be transformed to much simpler ones called "diagonal matrices".

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