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

aortiH 2020-10-31 Answered
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
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Expert Answer

Brighton
Answered 2020-11-01 Author has 103 answers
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})
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