Let T : V right arrow→ W be a linear transformation from a vector space V into a vector space W. Prove that the range of T is a subspace of W. [Hint: Typical elements of the range have the form T(x) and T(w) for some x and w in V.] What would be the most useful method of proving that the range of T is a subspace of W? A. Show that the three properties that define a subspace are satisfied by the vectors in the range of T. B. Show that there is a set of vectors that span the range of T. C. Show that the range of T must be all of W. D. Find a matrix such that the range of T is the column space of that matrix. E. Find a matrix such that the range of T is the null space of that matrix. F. Show that all ten axioms for a vector space are satisfied by the vectors in the range of T.

Let T​ : V right arrow→ W be a linear transformation from a vector space V into a vector space W. Prove that the range of T is a subspace of W.​ [Hint: Typical elements of the range have the form ​T(x​) and ​T(w​) for some x and w in​ V.] What would be the most useful method of proving that the range of T is a subspace of​ W? A. Show that the three properties that define a subspace are satisfied by the vectors in the range of T. B. Show that there is a set of vectors that span the range of T. C. Show that the range of T must be all of W. D. Find a matrix such that the range of T is the column space of that matrix. E. Find a matrix such that the range of T is the null space of that matrix. F. Show that all ten axioms for a vector space are satisfied by the vectors in the range of T.

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roots are from trees

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