Let over→{e1},over→{e2},over→{e3} be standard unit vectors along the coordinate axes in R3.Let S and T be the linear transformations defind in R3.Show that if S(over→{e}1)=T(over→{e}1),S(over→{e}2)=T(over→{e}2),S(over→{e}3)=T(over→{e}3) then S(over→{x})=T(over→{x}) for any over→{x}∈R3

Question

Let over→{e1},over→{e2},over→{e3} be standard unit vectors along the coordinate axes in R3.Let S and T be the linear transformations defind in R3.Show that if  S(over→{e}1)=T(over→{e}1),S(over→{e}2)=T(over→{e}2),S(over→{e}3)=T(over→{e}3)  then  S(over→{x})=T(over→{x})  for any over→{x}∈R3

Brighton · Accepted Answer

Any vector over→{x}∈R3can be written as, over→{x}=aover→{e}1 + bover→{e2} + cover→{e}3  Then, using properties of linear transformation,  ⇒S(over→{x})  ⇒S(aover→{e}1 + bover→{e}2 + cover→{e}3)  ⇒aS(over→{e}1) + bS(over→{e}2) + cS(over→{e}3)  We know that,  [S(over→{e}1)=T(over→{e}1),S(over→{e}2)=T(over→{e}2),S(over→{e}3)=T(over→{e}3)]  ⇒aT(over→{e}1) + bT(over→{e}2) + cT(over→{e}3)  ⇒T(a over→{e}1 + b over→{e}2 + c over→{e}3)  ⇒T(over→{x})  Thus, proved S(over→{x})=T(over→{x})

text{Let} overrightarrow{e_1}, overrightarrow{e_2}, overrightarrow{e_3} text{be standard unit vectors along the coordinate axes in} R^3. text{Let S an

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