Math 375 - 10/10

Linear Transformations • Definition ○ Let V and W be two vector spaces ○ Then a map/function/transformation/mapping ○ T:V→W is called linear if ○ {■8(T(x+y)=T(x)+T(y)&∀x,y∈V@T(c⋅x)=c⋅T(x)&∀x∈V,c∈R┤ • Mapping notation ○ In the mapping T:V→W ○ V is called domain ○ W is called codomain or target set ○ T(v) must be defined ∀v∈V ○ T(v) always belongs to W • Example 1 ○ Let V,W be any vector space ○ Define Tx=0, ∀x∈V ○ {█(T(x+y)=0@T(x)+T(y)=0+0=0)┤⇒T(x+y)=Tx+Ty ○ {█(T(c⋅x)=0@c⋅T(x)=c⋅0=0)┤⇒T(c⋅x)=c⋅T(x) ○ Therefore this mapping is a linear transformation • Example 2 ○ Let V,W be any vector space ○ Define Tv=w≠0, ∀v∈V ○ T(x)+T(y)=2w≠w=T(x+y) ○ Therefore this mapping is not a linear transformation • Example 3 ○ Let V=W be the same vector space ○ Define Tx=x, ∀v∈V ○ Then T is a linear transformation ○ T is called the identity map from V to V ○ Common notations: id, id_V, 1_V • Example 4 ○ Let V=W=R2 be the same vector space ○ Define T(x,y)=(2x,2y) ○ T(u)+T(v)=2u+2v=2(u+v)=T(u+v) ○ T(c⋅u)=2c⋅u=c⋅(2u)=c⋅T(u) ○ Therefore T is a linear transformation • Example 5 ○ Let V=W=R2 be the same vector space ○ Define T(a,b)=(b,a) ○ It s reflection in the diagonal • Example 6 ○ Let V=W=R2 be the same vector space ○ Define Tu=u rotated by 30° counter-clockwise ○ Proof by graph T(u+v)=T(u)+T(v) ○ We can also prove that T(c⋅v)=c⋅T(v) ○ Therefore T is a linear transformation Linear Transformation on Basis • Theorem ○ Suppose T:V→W is a linear transformation ○ Let {e_1,…,e_n } be a basis for V ○ Then T is completely defermined by ○ {Te_1,,Te_2…,Te_n } ○ Suppose we known Te_1,Te_2,…,Te_n, ○ and let x∈V be given ○ Then there are c_1,c_2,…,c_n∈R ○ such that x=c_1 e_1+c_2 e_2+…+c_n e_n, then ○ T(x)=T(c_1 e_1+c_2 e_2+…+c_n e_n ) ○ =T(c_1 e_1 )+T(c_2 e_2 )+…+T(c_n e_n ) ○ =c_1 Te_1+c_2 Te_2+…c_n Te_n • Example (Rotation) ○ Let V=W=R2 be the same vector space ○ Define T rotate by θ counter-clockwise ○ Pick a basis {e_1,e_2 }, where § e_1=(█(1@0)) § e_2=(█(0@1)) ○ Compute Te_1, Te_2 § Te_1=(█(cos⁡θ@sin⁡θ )) § Te_2=(█(−sin⁡θ@cos⁡θ )) ○ Compute T(ae_1+be_2 ) § T(ae_1+be_2 ) § =aTe_1+bTe_2 § =a(█(cos⁡θ@sin⁡θ ))+b(█(−sin⁡θ@cos⁡θ )) § =(█(a cos⁡θ−b sin⁡θ@a sin⁡θ+b cos⁡θ )) Solving System of Equations • Setup ○ {█(a_11 x_1+a_12 x_2+…+a_1n x_n=y_1@a_21 x_1+a_22 x_2+…+a_2n x_n=y_2@⋮@a_n1 x_1+a_n2 x_2+…+a_nn x_n=y_n )┤ ○ Define a transformation T: Rn→Rn ○ Let x=(█(x_1@⋮@x_n )), y=(█(y_1@⋮@y_n )) ○ Then Tx=y is a linear transformation • Property of one-to-one map ○ A linear map T:V→W is a one-to-one map ○ if for all u,v∈V ○ Tu=Tv⇒u=v ○ i.e. The equation Tx=y has at most one solution • Example of one-to-one map ○ Let V=R2, W=R3 ○ T(x_1,x_2 )=(x_1,x_2,0)=(y_1,y_2,y_3 ) ○ {█(1x_1+0x_2=y_1@0x_1+1x_2=y_2@0x_1+0x_2=y_3 )┤⇒{█(y_1=x_1@y_2=x_2@y_3=0)┤ ○ Three equations, two unknowns • Theorem ○ A linear map T:V→W is injective ○ if for all x∈V ○ Tx=0⇒x=0