using basis of vwe can thus represent each vector as row or column of numbers [ an or ( an standard basis transpose typicallyvectors in rn are represented via the standard basisconsisting of unit vectors (pointse ( )en ( as consequenceany point ( xn rn can be representedusing the standard basisas row or column vector of the form ( abovewith ai xi we will also write [ xn ]for the corresponding column vectorwhere denotes the transpose to avoid confusionwe will use the convention from now on that generic vector is always represented via the standard basis as column vector the corresponding row vector is denoted by xexample (linear transformationtake the matrix - - it transforms the two basis vectors [ ]and [ ]shown in red and blue in the left panel of figure to the vectors [ - ]and [ - ]shown on the right panel similarlythe points on the unit circle are transformed to an ellipse rank matrix can be viewed as an array of rows and columns that defines linear transformation from rn to rm (or for complex matricesfrom cn to cm the matrix is said to be square if if an are the columns of athat isa [ an ]and if [ xn ]then ax xn an in particularthe standard basis vector ek is mapped to the vector ak we sometimes use the notation [ai ]to denote matrix whose (ij)-th element is ai when we wish to emphasize that matrix is real-valued with rows and columnswe write rmxn the rank of matrix is the number of linearly independent rows orequivalentlythe number of linearly independent columns matrix linear transformation - - - - - - - - figure linear transformation of the unit circle