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Social ScienceThere exists a linear transformation T:IR^2019->IR^2019 that is onto but not one-to-one. Is this true or false?
There exists a linear transformation T:IR^2019->IR^2019 that is onto but not one-to-one. Is this true or false?
Expert Answer
To determine whether the given statement about linear transformations is true or not
The number 2019 is immaterial. We will prove that the statement is FALSE. (not only fo r2019, but for any positive integer n).
Let n be a positive integer and V be a finite dimensional vector space. Then T is a linear transfor...
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Theorem T V>V,onto,linear, with dim()n, finite. Then T is (also) one -to-one Proof Let vi=1,2,..n} be a basis for V. As T is onto,3{u, i= 1,2,..n},with T(u) v Claim: i-1,2.n} is a basis for V. Enough to prove linear independence. Now cu,0Tcu) =T(0) = 0(by linearity) - ΣςΤα) = 0(ν linearip) e0(by linearity) c0,Vi, as v, are linearly independent.
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