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Question: 30 please...

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0 0 2 3 0 0 0 1 0o 0 0 -1 0 0 0 01 1 0 0 4 0 1 0 0 0 0 01 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0 00 1 0 100 0 I Ten] is unique: if J. o 2 0 1 27, Prove directly that the standard matrix A = [T)-[1で1) l TC2) I B is another matrix such that TG)-Ax-Br for all x e R,, then A = B Ar = Br into (A-B)x which would make (A B) a non-zero vector. [T]. Hint: rewrite R 0m. If any of the entries ofA-B is non-zero, think of a specific x 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 permutation matrix, because it is a rearrangement of the columns of I Find the standard matrix of the scaling operator Sk on R, where k e R, given by S,G)- 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: TG)- any linear transformation T : R-R, we must have: T(0) = 0m 29. 30. Use this to prove that for 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(,) = 0m. Hint: compute O+) in two different ways. 32. Now, using the Homogeneity Property and the multiplicative property of the scalar zero, prove How directly that for any linear transformation T: R Rm, we must have T O should the Hint from the previous Exercise be modified? 169 Section 2.1 Mapping Spaces: Introduction to Linear Transformations 30 please
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