I need help with this textbook problem Solution kernel and range are the words r

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I need help with this textbook problem

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kernel and range are the words refers linear transformation in the topic linear algebra suppose U and V are two vector spaces defined over the same field F. T:U--> V is a function such that T(au+bv) = aT(u) +bT(v) for every u, v in U and a, b in F, then T is said to be the linear transformation. a linear transformation is a vector space homomorphism but preserves the linearity. R(T) is the range of the linear transformation T is a subspace in V. kernel T is a subspace of U. the kernel of the linear transformation is called the null space of T. we have the connecting theorem on the dimensions of the vector spaces and subspaces. if U and V are finite dimensional vector spaces and T is a linear transformation from U to V, then dimension of R(T) + dimension of N(T) = dimension of U. for any more theorems or clarifications and how linear algebra is connected to matrices can be made simple provided you ask them straright away. thank you in waiting.

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