Synonyms: Linear Transformation, Linear Operator, Linear map, Vector Space Homomorphism, If Q: V → W is a linear mapping over a field F, then for any scalars a i in F A linear mappingįor all vectors v 1, v 2 ε V and scalars a, b ε F.Ī generalization of this definition gives the following basic property of linear mappings:īasic property. Let V and W be vector spaces over the same field F. For any a in F and any x in V, Q(ax) = aQ(x)īy combining the above two conditions of linearity we obtain a third definition:Īlternate definition 2. For any x,y in V, Q(x + y) = Q(x) + Q(y)Ģ. Is a mapping Q: V → W that satisfies the following two conditions:ġ. There are alternative but equivalent definitions of a linear mapping.Īlternate definition 1. This represents a mapping of n-space into Where x and y are n-vectors and A is an nxn matrix. Linearity of a mapping is that subspaces are mapped into subspaces.Īn example of such a mapping in n-dimensional space, V n(F), is If it is nonsingular it effects a vector space isomorphism. Vector addition and that of scalar multiplication. Numbers a and b may be real, complex or of any field for which multiplication with elements ofĪ linear mapping is a mapping that 'preserves' the two basic operations of a vector space, that of Takes ax + by into ax' + by' for all a and b if it takes vectors x and y in V into x' and y' in W. V and W be vector spaces over the same field F. Transformation) is a mapping defined on a vector space that is linear in the following sense: Let Linear mapping (or linear transformation). L INEAR MAPPING (OR LINEAR TRANSFORMATION)ĭef. Linear mapping, Linear transformation, Linear operator, Linear map,

Onto vs one to one vector