Chapter 2 Linear Transformations Linear Transformations Loosely speaking, a linear transformation is a function from one vector space to another that preserves the vector space operations. Let us be more precise. Definition Let => and be vector spaces over a field -. A function ¢=¦> is a linear transformation if ²" b #³ ~ ²"³ b ²#³ for all scalars Á - and vectors ", # = . A linear transformation ¢ = ¦ = is called a linear operator on ==. The set of all linear transformations from to >²=Á>³ is denoted by B and the set of all linear operators on = is denoted by B²= ³. We should mention that some authors use the term linear operator for any linear transformation from => to . Definition The following terms are also employed: 1) homomorphism for linear transformation 2) endomorphism for linear operator 3) monomorphism (or embedding) for injective linear transformation 4) epimorphism for surjective linear transformation 5) isomorphism for bijective linear transformation. 6) automorphism for bijective linear operator. Example 2.1 1) The derivative +¢ = ¦ = is a linear operator on the vector space = of all infinitely differentiable functions on s. 56 Advanced Linear Algebra 2) The integral operator ¢-´%µ¦-´%µ defined by % ²³ ~ ²!³! is a linear operator on -´%µ. 3) Let (d- be an matrix over . The function ( ¢-¦- defined by (²#³ ~ (#, where all vectors are written as column vectors, is a linear transformation from -- to . This function is just multiplication by (. 4) The coordinate map ¢= ¦- of an -dimensional vector space is a linear transformation from =- to . The set BB²= Á > ³ is a vector space in its own right and ²= ³ has the structure of an algebra, as defined in Chapter 0. Theorem 2.1 1) The set B²= Á > ³ is a vector space under ordinary addition of functions and scalar multiplication of functions by elements of - . 2) If B²<Á=³ and B ²=Á>³ then the composition B is in ²<Á>³. 3) If B²=Á>³ is bijective then c ²>Á=³ B . 4) The vector space B²= ³ is an algebra, where multiplication is composition of functions. The identity map B²=³ is the multiplicative identity and the zero map B ²= ³ is the additive identity. Proof. We prove only part 3). Let ¢= ¦> be a bijective linear transformation. Then c¢> ¦= is a well-defined function and since any two vectors $ and $ in > have the form $ ~ ²# ³ and $ ~ ²# ³, we have c c ²$ b$³~ ² ²#³b ²#³³ c ~ ² ²# b # ³³ ~# b# c c ~ ²$³b ²$³ which shows that c is linear. One of the easiest ways to define a linear transformation is to give its values on a basis. The following theorem says that we may assign these values arbitrarily and obtain a unique linear transformation by linear extension to the entire domain. Theorem 2.2 Let => and be vector spaces and let 8 ~¸#0¹ be a basis for =. Then we can define a linear transformation B ²= Á > ³ by specifying the values of 8²# ³ > arbitrarily for all # and extending the domain of to = using linearity, that is, ²# bÄb # ³~ ²#³bÄb ²#³ Linear Transformations 57 This process uniquely defines a linear transformation, that is, if Á B ²=Á>³ satisfy ²#³~ ²#³ for all # 8 then ~ . Proof. The crucial point is that the extension by linearity is well-defined, since each vector in = has a unique representation as a linear combination of a finite number of vectors in 8. We leave the details to the reader. Note that if B²=Á>³ and if : is a subspace of =, then the restriction O: of to ::> is a linear transformation from to . The Kernel and Image of a Linear Transformation There are two very important vector spaces associated with a linear transformation from => to . Definition Let B²=Á>³. The subspace ker²³~¸#= ²#³~¹ is called the kernel of and the subspace im²³~¸²#³#=¹ is called the image of . The dimension of ker²³ is called the nullity of and is denoted by null²³. The dimension of im ²³ is called the rank of and is denoted by rk²³ . It is routine to show that ker²³ is a subspace of = and im ²³ is a subspace of > . Moreover, we have the following. Theorem 2.3 Let B²=Á>³. Then 1) is surjective if and only if im²³~> 2) is injective if and only if ker² ³ ~ ¸¹ Proof. The first statement is merely a restatement of the definition of surjectivity. To see the validity of the second statement, observe that ²"³ ~ ²#³ ¯ ²" c #³ ~ ¯ " c # ker ² ³ Hence, if ker² ³ ~ ¸¹ then ²"³ ~ ²#³ ¯ " ~ #, which shows that is injective. Conversely, if is injective and " ker ² ³ then ²"³ ~ ²³ and so " ~ . This shows that ker ² ³ ~ ¸¹. Isomorphisms Definition A bijective linear transformation ¢= ¦> is called an isomorphism from => to . When an isomorphism from => to exists, we say that => and are isomorphic and write =>. Example 2.2 Let dim²= ³ ~ . For any ordered basis 8 of = , the coordinate map 8¢= ¦- that sends each vector #= to its coordinate matrix 58 Advanced Linear Algebra ´#µ8 - is an isomorphism. Hence, any -dimensional vector space over - is isomorphic to - . Isomorphic vector spaces share many properties, as the next theorem shows. If B²=Á>³ and := we write ²:³ ~ ¸ ² ³ :¹ Theorem 2.4 Let B²=Á>³ be an isomorphism. Let :=. Then 1) := spans if and only if ²:³> spans . 2) : is linearly independent in = if and only if ²:³ is linearly independent in > . 3) := is a basis for if and only if ²:³> is a basis for . An isomorphism can be characterized as a linear transformation ¢= ¦> that maps a basis for => to a basis for . Theorem 2.5 A linear transformation B²=Á>³ is an isomorphism if and only if there is a basis 88 of =²³> for which is a basis of . In this case, maps any basis of => to a basis of . The following theorem says that, up to isomorphism, there is only one vector space of any given dimension. Theorem 2.6 Let => and be vector spaces over -. Then => if and only if dim²= ³ ~ dim ²> ³. In Example 2.2, we saw that any -dimensional vector space is isomorphic to ) -. Now suppose that ) is a set of cardinality and let ²- ³ be the vector space of all functions from )- to with finite support. We leave it to the reader ) to show that the functions ²- ³ defined for all ), by %~if ²%³ ~ F %£if )) form a basis for ²- ³, called the standard basis. Hence, dim ²²- ³ ³ ~(( ) . It follows that for any cardinal number , there is a vector space of dimension . ) Also, any vector space of dimension is isomorphic to ²- ³. Theorem 2.7 If is a natural number then any -dimensional vector space over -- is isomorphic to . If is any cardinal number and if ) is a set of cardinality then any -dimensional vector space over - is isomorphic to the ) vector space ²- ³ of all functions from ) to - with finite support. Linear Transformations 59 The Rank Plus Nullity Theorem Let B²=Á>³. Since any subspace of = has a complement, we can write =~ker ²³l ker ²³ where ker²³ is a complement of ker ²³ in =. It follows that dim²= ³ ~ dim ² ker ² ³³ b dim ² ker ² ³ ³ Now, the restriction of to ker²³ ¢²³¦>ker is injective, since ker² ³ ~ ker ² ³ q ker ² ³ ~ ¸¹ Also, im²³²³ im. For the reverse inclusion, if ²#³²³ im then since #~"b$"²³$²³ for ker and ker , we have ²#³ ~ ²"³ b ²$³ ~ ²$³ ~ ²$³ im ² ³ Thus im²³~²³ im . It follows that ker²³ im ²³ From this, we deduce the following theorem. Theorem 2.8 Let B²=Á>³. 1) Any complement of ker²³ is isomorphic to im ²³ 2) ()The rank plus nullity theorem dim² ker ² ³³ b dim ²im ² ³³ ~ dim ²= ³ or, in other notation, rk² ³ b null ² ³ ~dim ²= ³ Theorem 2.8 has an important corollary. Corollary 2.9 Let B ²=Á>³, where dim ²=³~ dim ²>³B. Then is injective if and only if it is surjective. Note that this result fails if the vector spaces are not finite-dimensional. Linear Transformations from -- to Recall that for any d matrix ( over - the multiplication map (²#³ ~ (# 60 Advanced Linear Algebra is a linear transformation. In fact, any linear transformation B²-Á-³ has this form, that is, is just multiplication by a matrix, for we have ²³ 2323² ³ Ä ² ³ ~ ² ³ Ä ² ³ ~ ² ³ and so ~ ( where ( ~23 ² ³ Ä ² ³ Theorem 2.10 1) If ( is an d matrix over - then B( ²- Á - ³. 2) If B²-Á-³ then ~( where (~² ²³Ä ²³³ The matrix ( is called the matrix of . Example 2.3 Consider the linear transformation ¢- ¦- defined by ²%Á&Á'³~²%c&Á'Á%b&b'³ Then we have, in column form vy% v %c& y v c% yvy &~~ ' & wz' w %b&b' z w zwz ' and so the standard matrix of is vyc (~ wz If (CÁ then since the image of ( is the column space of (, we have dim²²³³b²(³~²-³ ker( rk dim This gives the following useful result. Theorem 2.11 Let (d- be an matrix over . 1) (¢- ¦- is injective if and only if rk ²(³~n. 2) (¢- ¦- is surjective if and only if rk ²(³~m. Change of Basis Matrices Suppose that 89~²ÁÃÁ ³ and ~²ÁÃÁ ³ are ordered bases for a vector space =´#µ´#µ. It is natural to ask how the coordinate matrices 89 and are c related. The map that takes ´#µ89899 to ´#µ is Á ~ 8 and is called the change of basis operator (or change of coordinates operator). Since 89Á is an operator on - , it has the form ( where Linear Transformations 61 (~²89Á ²³ÁÃÁ 89 Á ²³³ c c ~ ²989888 ²´ µ ³Á à Á ²´ µ ³³ ~ ²´ µ99 Á à Á ´ µ ³³ We denote (4 by 89, and call it the change of basis matrix from 89 to . Theorem 2.12 Let 89~²ÁÃÁ ³ and be ordered bases for a vector space c =~-. Then the change of basis operator 89Á 9 8 is an automorphism of , whose standard matrix is 489, ~²´ µ 9 ÁÃÁ´ µ 9 ³³ Hence ´#µ9898 ~ 4Á ´#µ c and 4~498Á 89, . Consider the equation (~489Á or equivalently, (~²´ µ99 ÁÃÁ´ µ ³³ Then given any two of (dÁ (an invertible matrix)8 (an ordered basis for --) and 9 (an order basis for ), the third component is uniquely determined by this equation.