theorems about LINEAR MAPPINGS

T is a linear mapping from a vector space V into a vector space W U x x T y y T the range the domain

T is a linear mapping from a vector space V into a vector space W U x x T y y T x T y + x y + the range x y + T( ) = T the domain

THEOREM 1 If T is a linear mapping from V into W then T( ) = V W

T is a linear mapping from a vector space V into a vector space W x T U V W the range T( ) = V W the domain

T( ) T( ) T( ) + x = x + x = linear mapping V T( ) + V x = linear mapping must be the identity in the range

THEOREM 2 If T is a linear mapping from V into W then T is one to one if and only if Null Space of T = { 0 }

T is a linear mapping from a vector space V into a vector space W x T null space U V W If T is one to one the range the null space contains only zero. the domain

T is a linear mapping from a vector space V into a vector space W If the null space contains only zero. x T null space U and = x T y then - = 0 x T y V W linear x y - x then T( - ) = 0 y x the range y x y - then is in the null space x y - then = 0 the domain x y = then

T is a linear mapping from a vector space V into a vector space W If the null space contains only zero. x T null space U and = x T y then - = 0 x T y V W linear x y - x then T( - ) = 0 y x y the range x y - then is in the null space x y - then = 0 the domain x y = then

T is a linear mapping from a vector space V into a vector space W If the null space contains only zero. x T null space U and = x T y then - = 0 x T y V W linear x y - x then T( - ) = 0 y x y y the range x y - then is in the null space x y - then = 0 the domain x y = then

THEOREM 3 If T is a linear mapping from V into W then The Null Space of T is a vector space

T is a linear mapping from a vector space V into a vector space W Let a and b be any two vectors in the null space V W a T( ) = 0 b T( ) = 0 x T null space U a T( ) + b T( ) = 0 a b W a T( b + ) = 0 a + T is linear the range b + is in the nullspace a the domain

THEOREM 4 If T is a linear mapping from V into W then The Range of T is a vector space

T is a linear mapping from a vector space V into a vector space W x T y and are in the range of T V W x T x y and are in the domain null space U x x T x + y y T x y V W y x + y ) T( x + y is in the domain the range x + y ) is in the range T( T is linear the domain x T y + is in the range of T

THEOREM 5 If T is a linear mapping from V into W then the dimension of the DOMAIN = the dimension of the RANGE + the dimension of the NULLSPACE

T is a linear mapping from a vector space V into a vector space W is a basis for the null space V W is a basis for the domain x T null space U x W y the range the domain is a basis for the range