How to know if a linear transformation is invertible?
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Let VV and WW be vector spaces over the field FFhaving the same finite dimension. Let T:V→WT:V→Wbe a linear transformation.
TT is said to be invertible if there is a linear transformation S:W→VS:W→V such that S(T(x))=xS(T(x))=x for all x∈Vx∈V.SS is called the inverse of TT. In casual terms, SSundoes whatever TT does to an input xx.
In fact, under the assumptions at the beginning, TT is invertible if and only if TT is bijective. Here, we give a proof that bijectivity implies invertibility. The other direction is left as an exercise.
Suppose that TT is bijective. So for each x∈Vx∈V, there is a unique vector in WW, call it yxyx such that T(x)=yxT(x)=yx. Define S:W→VS:W→V as follows: for each y∈Wy∈W, let x∈Vx∈V be such that y=yxy=yx. Note that such an xxexists as TT is surjective and the choice is unique as TT is injective. Hence, SS is a well-defined function from WW to VV and it satisfies S(T(x))
please add brainlist to my answer
Let VV and WW be vector spaces over the field FFhaving the same finite dimension. Let T:V→WT:V→Wbe a linear transformation.
TT is said to be invertible if there is a linear transformation S:W→VS:W→V such that S(T(x))=xS(T(x))=x for all x∈Vx∈V.SS is called the inverse of TT. In casual terms, SSundoes whatever TT does to an input xx.
In fact, under the assumptions at the beginning, TT is invertible if and only if TT is bijective. Here, we give a proof that bijectivity implies invertibility. The other direction is left as an exercise.
Suppose that TT is bijective. So for each x∈Vx∈V, there is a unique vector in WW, call it yxyx such that T(x)=yxT(x)=yx. Define S:W→VS:W→V as follows: for each y∈Wy∈W, let x∈Vx∈V be such that y=yxy=yx. Note that such an xxexists as TT is surjective and the choice is unique as TT is injective. Hence, SS is a well-defined function from WW to VV and it satisfies S(T(x))
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