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Question: 32 please...

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0 0 0 1 27, Prove directly that the standard matrix A-[7]-[7G) İ TG2) I I Ten] is unique: if B is another matrix such that TG)-Ax-Br for all x e R, then A = B = [T]. Hint: rewrite ABi into (-B-0If any of the entries of A - B is non-zero, think of a specific ž e R which would make (A -B) a non-zero vector. 28. Show that the matrix of T from Exercise 10 can be obtained from the identity matrix Is by a sequence of Type 2 row operations. For this reason, this is an example of what is called a mutation matrir, because it is a rearrangement of the columns of I -a. 29. Find the standard matrix of the scaling operator S on R, where k e R, given by S C) 30. Starting with the two properties of a linear transformation T, we found from our only Theorem in this Section that T can be computed using a matrix product: TC) Ax. Use this to prove any linear transformation T : R → R, we must have: T( 0 ) = 0 31. Now, using the Additivity Property and the property of the zero vector, prove directly that for any linear transformation T: R → Rm, we must have T(0,) = 0m. Hint: compute T(O, +O,) in two different ways. Now, using the Homogeneity Property and the multiplicative property of the scalar zero, prove directly that for any linear transformation T : R-R, we must have T(0) =0m. How should the Hint from the previous Exercise be modified? 32. 169 Section 2.1 Mapping Spaces: Introduction to Linear Transformations 32 please
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