5: (6 Points) Let T : Rn → Rm b e a linear transformation. Prove T(%)...
Define the linear transformation T: Rn → Rm by πν) = Av. Find the dimensions of Rn and Rm. A=12-2 24-2 1 dimension of R dimension of Rm
Q8 6 Points Let T : R2 + Rº be a linear transformation with PT(x) = x2 – 1. Decide whether or not such a T is always diagonalizable. Justify your answer.. Q8.2 3 Points Determine/Compute the linear transformation T2 : R2 + R2, VH T(T(u)).
6. Let T P2 P be a linear transformation such that T P2P2 is still a linear trans formation such that T(1) 2r22 T(2-)=2 T(1) = 2r22 T(12 - )=2 T(x2x= 2r T(r2)2x (a) (6 points) Find the matrix for T in some basis B. Specify the basis that you use. (d) (4 points) Find a basis for the eigenspace E2. (b) (2 points) Find det(T) and tr(T') (e) (4 points) Find a basis = (f,9,h) for P2 such that...
Let U be an open subset of R. Let f: U C Rn → Rm. (a) Prove that f is continuously differentiable if and only if for each a є U, for each E > 0, there exists δ > 0 such that for each x E U, if IIx-all < δ, then llDf(x)-Df(a) ll < ε. (b) Let m n. Prove that if f is continuously differentiable, a E U, and Df (a) is invertible, then there exists δ...
1. (a) Let T:R' R'be defined by T(x) = 5 -2. Is T a linear transformation? If so, prove that it is. If not, explain why not. (b) More generally than part (a), suppose that T:R → R is defined by T(x) = ax +b, where a and b are constants. What must be true about a and b in order for T to be a linear transformation? Explain your answer.
Let TRm → Rn be a linear transformation, and let p be a vector and S a set in R Show that the image of p + S under T is the translated set T(p) + T(S) n R What would be the first step in translating p+ S? OA. Rewrite p+ S so that it does not use sets. O B. Rewrite p+S so that it does not use vectors O c. Rewrite p + S as a difference...
SF78. Consider the linear map T : Rn → Rm defined by T(v) = Av where A=12 43 6 12-7 (a) What is m? (b) What is n? (c) The image of T is a subspace of R. What is i? (d) The image of T is isomorphic to R. What is j? e The kermel of T is isomorphic to Rt. What is k7 (f) The kernel of T is a subspace of R. What is ?
2. (5 points) Let T: R2 + R3 be a linear transformation with 2x1 - x2] 1-3x1 + x2 | 2x1 – 3x2 Find x = (x) <R? such that [0] -1 T(x) = (-4)
a Suppose {יוע ע "ע 3is a linearly dependent set in R. Let T : Rn Rm be a linear transformation. Explain why { T(L| ), Tjas), Γ(L3 ) must be linearly dependent in Rin. b Suppose (x, t.,) is a linearly independent set in R". Let Tbe a linear transformation. Do T(Li ) , T(L, ), T(ש ) and explain why not. have to be linearly independent in R ? Explain why, or give a counterexample
11.) Let T:R" - R"be a linear transformation. Prove T is onto if and only if T is one-to-one. 12.) Let T:R" - R" and S:R" - R" be linear transformations such that TSX=X for all x ER". Find an example such that ST(x))+x for some xER". - .-.n that tidul,