Subspaces, Linear In dependence, Matrix Inverse and Bases

A subset of Rn or C is called a subspace if it is closed under vector addition and vector scaling. Typical subspaces are (1) the set of all linear combinations of a number of vectors 

An important application of the theory of linear equations is to decide which vectors among a given set of subspace generating vectors are essential for the given subspace, and which can replicate the action of others and therefore are not needed. 

Such minimal spanning sets of vectors for a given subspace are called a basis for the subspace. The vectors of a basis for a subspace are also a maximally linearly independent set of vectors. Here we call a set of k column vectors in Rn linearly independent if 

As matrices represent linear mappings between finite dimensional vector spaces we are interested to find out which linear mappings can be inverted, i.e., for which matrices Am n Rn — RTO does there exist an inverse matrix A Rm — R with A-1 (Ax) = x for all x Rn, and how can we find A 1 from A if possible. 

The second property (ii) is essential for / 1 to be a function in the unique assignment sense. 

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C) Subspaces, Linear (In)dependence, Matrix Inverse and Bases... 

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