Linear transformation with special propertiesLinear transformation f:R10→R7 has an attribute that every vector v for which is true that f(v)=0 is in linear span (1,2,…,10)T,(1,1…,1)T&gt;. Create such transformation or prove that it doesn't exists.

Question

Linear transformation with special propertiesLinear transformation f:R10→R7 has an attribute that every vector v for which is true that f(v)=0 is in linear span (1,2,…,10)T,(1,1…,1)T&amp;gt;. Create such transformation or prove that it doesn&#039;t exists.

diascordicm5 · Accepted Answer

The linear transformation should satisfy that  7≥dim⁡(Im(f))=dim⁡R10−dim⁡(ker(f))=10−dim  (ker(f))  Therefore dim⁡(ker(f))≥3.

prudomiajpt · Accepted Answer

there is no such transformation T given that ker(T)⊂span{(1,2,⋯,10)T,(1,1,⋯,1)T} because the dimension row space of T is the same as the dimension of the column space which is less or equal to seven. by the nullity theorem, dimension of ker(T) is greater or equal to three. and that contradicts that the dimension of ker(T) is two.

Linear transformation with special properties Linear transformation f:R^{10}\rightarrow R^{7} has an

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