Consider the bases B = {u₁, u2} and B' = {v1, v2} for R², where 21= 8 U2 = V₁ = v₂ = 6 8 Find the transition matrix from B' to B. The transition matrix from B' to B is P = (H)

Consider a linear transformation T from R2 to R² for which (:) -}- T and T Find the matrix A of T. A =

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Practice Another Find the transition matrix from B to B'. B = {(-1, 0, 0), (0, 1, 0), (0, 0, -1)}, B' = {(0, 0, 9), (1, 3, 0), (8, 0, 9)}

Find an orthogonal diagonalization for A = [[1, 1,0], [1, 1, 1], [0, 1, 1]], i. e. find an orthogonal matrix U and a diagonal matrix D such that U^ (TT)AU = D. Any empty entries are assumed to be 0.

Let J, be then X n matrix whose entries are all equal to 1, and let D(^1, . , An) be the n X n diagonal matrix whose non-zero entries are 11, ... , An E R. Let x = (x1, ... , Xn) be a row vector in R' and let x' be its transpose, a column vector. 1. Show that xJ,x' > 0, for all x1, ... , Xn E R. 2. Assume that the scalars 11, ..., An are all strictly positive. Show that x D(11, ... , An)x' is strictly positive for all (x1, ..., Xn), unless x1 = •… Xn = 0. 3. Use questions 2 and 1 above to show that D(^1, .. , An) + 1Jn is an invertible matrix when 11, , ^n and 1 are all strictly positive.

U₁ = -D₁ --- --- -- = = V2 = (a) Find the transition matrix from the basis E = {v1, v2} to the basis F = {u₁, u₂} of R². (b) Use the transition matrix from (a) to write x = 5 from F. -4 [5] as a linear combination of the vectors

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