Let T : R → R° be a linear transformation.Let v,, v,, V3 be nonzero vectors in R’ such that:(T – 21)v,(T – 21)v,(T – 2I)v; = v2Prove that:(1) 1 = 2 is the only eigenvalue for T.(2) T is not diagonalizable.(3) Find a basis B for R' such that T(2 1 0`0 2 10 0 2,В

1) Given that v1, v2, v3 be nonzero vectors in ℝ3 such that T-2Iv1=0T-2Iv2=v1T-2Iv3=v2 That is,…

Let T : R → R° be a linear transformation. Let v,, v2, v3 be nonzero vectors in R' such that: (T – 21)v, = 0, (T – 21)v, (T – 2I)v3 = v2 %3| Prove that: (1) A = 2 is the only eigenvalue for T. (2) T is not diagonalizable. (3) Find a basis B for R such that T (2 1 0\ 0 2 1 0 0 2 B ||

Let T : R → R° be a linear transformation. Let v,, v2, v3 be nonzero vectors in R' such that: (T – 21)v, = 0, (T – 21)v, (T – 2I)v3 = v2 %3| Prove that: (1) A = 2 is the only eigenvalue for T. (2) T is not diagonalizable. (3) Find a basis B for R such that T (2 1 0\ 0 2 1 0 0 2 B ||

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Consider the linear transformation T: R → Rn whose matrix A relative to the standard basis is given. 3 1 -[-28] 6 A = (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) (1₁, 1₂) = ( [ (b) Find a basis for each of the corresponding eigenspaces. B₁ = B₂ = { (c) Find the matrix A' for T relative to the basis B', where B' is made up of the basis vectors found in part (b). A' =
1) Find the transition matrix P from the basis B={ (-1,2). (2, 1)} of R* to the basis {(4, 3). (-3,2). Ifliul, . find (u),
4. Let W be the xy-plane in R³. (a) Find an orthonormal basis ū₁, 2 for W. (b) Construct the matrix Q = [ ₁ ü₂ ] and calculate the matrix QQT. (c) Consider the linear transformation T: R³ → R³ that projects each vector orthogonally onto the xy-plane. Note that T() = QQT. Use the matrix to calculate T(₁), T(2), and T(ē3). (d) Explain how your answers make sense geometrically.
If the matrix of change of basis form the basis B to the basis B' is A = (³) O A. OB. 04 (3) 00 (12) 52 21 then the first column of the matrix of change of basis from B to B is:
Find the coordinate matrix of x relative to the orthonormal basis B in R". x = (10, 20, 5), B = {G) (-), (0, o, 1)} [x]g = 1 1
5. Find the standard matrix A and A' for T = T2° T1 and T' = T1° T2, where T1:R2 → R3, T1(x,y) = (x,x+y,y) and T2:R3 → R², T2(x,y,z) = (0,y). Use standard basis vectors to derive your re- sults.
The vectors -6 01 = 02 = 03 =| 4 form a basis for R' if and only if k +
Consider the linear transformation T: R^ →R" whose matrix A relative to the standard basis is given. [³ A = (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) (1₁, 12) = 4,5 B₂ 3-2 (b) Find a basis for each of the corresponding eigenspaces. B₁ = 1,2 128 1,1 = A' = 1 6 (c) Find the matrix A' for T relative to the basis B', where B' is made up of the basis vectors found in part (b). X 5 5 1 2 ↓ 1
3. Let W1 = (1, 1, 0, 1), w₂ = (−1, 1, 0, 0), w3 = = (1, 2,0,0). Let W = Span(w₁, W2, W3 (a) Use the Gram-Schmidt process to obtain an orthogonal basis {V₁, V2, V3} for W. (b) What is Projw (3, 3, 3, 3), the orthogonal projection of the vector (3,3,3,3) onto W?
7. (a) Consider the ordered basis ß = [1,1 +x, 1 + x + x²] for V = P₂ and F = R. Suppose is in P2 and you know these coordinates: [] = (1, 2, 3). Find the vector 7. (b) Let A be a 2 x 2 matrix and a be a real number. Prove that det (a A)=a² det(A)