(a) Show that for each vector v ER", there is a vector a in the columnspace of T and a vector b in the nullspace of T such that v = a +b. (b) Show that there is a basis {V₁,...,Vn} of R" and a number 0 ≤ k ≤n such that (e) For the matrix TV₁ TVK TVk+1 TV₂ = T= 2 Vk 0 0. 2 3 5 6 -2 -4 -5 find a basis for R³ and a number k with the properties above.

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