Linear Algebra: Step by Step Chapter 5

Posted on August 16, 2015 by Sumant Sumant

Definition 5.2 A transformation T:V \to W is called a linear transformation \iff for all vectors u and v in the vector space V and for any scalar k we have

T(u+v)=T(u)+T(v) (T preserves vector addition)

T(ku)=kT(u) (T preserves scalar multiplication)

Proposition 5.3 Let V and W be vector spaces and u and v be vectors in V. Let T:V \to W be a linear transformation, then we have the following:

T(0) =0 where 0 is the zero vector

T(-u)=-T(u)

T(u-v)=T(u)-T(v)

Proposition 5.4

Let V and W be vector spaces. Let T: V \to W be a linear tranformation and \{ u_1,u_2,u_3,..,u_n \} be vectors in the vector space V such that v =k_1u_1+k_2u_2+...+k_nu_n where k’s are scalars. This means that the vector v is the linear combination of the u_{subscript} vectors. Then T(v)=k_1T(u_1)+k_2T(u_2)+...+k_nT(u_n)

Definition 5.5 Let T:V \to W be a linear transform (map). The set of all vectors v in V that are transformed to the zero vector in W is called the kernel of T, denoted by ker(T). It is the set of vectors v in V such that T(v)=0

Proposition 5.6 Let T:V \to W be a linear transformation (mapping) between the vector spaces V and W. Then the kernel of T is a subspace of the vector space V.

Definition 5.7 Let T:V \to W be a linear transform. Then ker(T) is also called the null space of T and the dimension of ker(T) is called the nullity of T which is denoted by nullity(T)

Definition 5.8 Let T:V \to W be a linear transform. the range(image) of the linear transform is set of all the output vectors w in W for which there are input vectors v in V such that T(v)=w

Definition 5.9: range(T)= {T(v)|v in V}

Proposition 5.10 Let V and W be vector spaces and T: V\to W be a linear transformation. The range of image of the transformation T is a subspace of the arrival vector space W

Proposition 5.12 Let T:V \to W be a linear transformation from an n dimensional vector space V to a vector space W. Then rank(T)+nullity(T)=n

Proposition 5.13 Let T: R^n \to R^m be a linear transformation given by T(x)=Ax. Then range(T) is the column space of A

Inverse Linear Transformations

Definition 5.14 Let T:V \to W be a linear transform, u and v be in the domain V. The transform T is one-to-one \iff u \ne v \Rightarrow T(u) \ne T(v)

Definition 5.15 T is one-to-one \iff T(u) = T(v) implies u=v

Proposition 5.16 Let T: V to W be a linear transformation between the vector spaces V and W. Then T is one-to-one \iff ker(T)={O}

Definition 5.18 Let T:V \to W be a linear transform. The transform T is onto \iff for every w in the arrival vector space W there exists at least one v in the start vector space V such that w = T(v)

Proposition 5.19 A linear transformation T: V \to W is onto iff range(T)=W

Proposition 5.20 Let T: V \to W be a linear transformation. Then T is onto \iff rank(T) = dim(W)

Proposition 5.21 If T: V \to W is a linear transformation and dim(V) = dim(W) then T is a one-to-one transformation \iff T is onto.

Proposition 5.22 If T: V \to W is a linear transformation and dim(V)=dim(W) then T is both one-to-one and onto \iff ker(T)= {0}

Proposition 5.23 Let T:V \to W be a bijective linear transform. The inverse transformation T^{-1}: W \to V is defined as v = T^{-1}(w) \iff T(v)=w. If T has an inverse transform, we refer to T as being invertible. To be invertible, T must be both one-to-one and onto

Proposition 5.24 Let T: V \to W be a linear transform which is both one-to-one and onto. Then the inverse transform T^{-1}: W \to V is also linear.

Definition 5.25 If the linear transformation T: V \to W is invertible then we say that the vector spaces V and W are isomorphic. Such a transformation  T is called an isomorphism.

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