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Let T:R" → Rh be the ödentity map T(*)=² show that undor any coordmate system the...
x 1.9.9 wuestion map Assume that Tis a linear transformation. Find the standard matrix of T. unchanged) and then reflects points through the line x2 + x4 T:R-R, first performs a horizontal shear that transforms e, into ez + 14, (leaving AO (Type an integer or simplified fraction for each matrix element.)
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,
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.
LI), Let T be a linear map from R5 to R3 with i. Show that T must be surjective. ii. Show that there can be no such T from R to R2
1. Let T: R2 – R? be the map "reflection in the line y = x"—you may assume this T is linear, let Eº be the standard basis of R2 and let B be the basis given by B = a) On the graph below, draw a line (colored if possible) joining each of the points each of the points (-). (). (1) and () woits image to its image under the map T. y = x b) Find the...
4. Let G : P(R) → P2(R) be a linear map given by G(u)(x) = (x + 1)u'r) - ur). Is G diagonalizable? If it is, find a basis of P(R) in which G is represented by a diagonal matrix 5. Let V = P2(C). Show that the operator (.) given by (u, v) = u(0) v(0) + u(1) v(1) + u(2) v(2) Vu, v E V is an inner product on V.
Let T: Rr - be a linear operator such that ToT Id Show that there is a basis B &Trelative to the basis B {ui , , , , , щ, vı , . . . ,VJofR" such that the representing matrix T Ul,. .. ,ur, Vi, has the form wherer +s-n(r or smay be zero), ie., adiagonal matrix whose diagonal entries are all
Let T: Rr - be a linear operator such that ToT Id Show that there is...
Question 21 Let T:R+ R2 be the transformation of rotation counterclockwise by 30 degrees, and let A be the standard matrix for T. Which one of the following statements must be true about A? A has one real eigenvalue with algebraic multiplicity 2. A has two distinct real eigenvalues. A has one real eigenvalue and one complex eigenvalue. A has two distinct complex eigenvalues. Question 22 Suppose A is a positive stochastic 2 x 2 matrix and A As n...
12. Show that T:R → R given by T(2) = + 1 is an isomorphism.
Let A= and 6 = Define the linear transformation T:R? +R by T'(X) = Ai. Find a vector # whose image under T' is 6. Is the vector i unique choose choose unique Submit answer not unique