(1) Let B = {√₁, 72, 73} be the basis of R³ defined by 1 *-(). *- (). *-(1) ), = = = Find the dual basis of B.
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- Find the coordinate vector of [4, -2, 1] relative to the ordered basis B= ([0, 1, 1], [2, 0, 0], [0, 3, 0]) of R^3.Verify that the polynomials 2 – x², x' – x, 2 – 3x² and 3 – x' form a basis for P3(x). Express cach of (i) 1 + x and (ii) x + x² as a linear combination of these basis vectors. 8.Find an algebraic description of the range of A by row reducing A^T and using the relation between R(A) and Row(A^T ) in order to obtain a basis of R(A).
- The vectors (attached) form a basis of R^3 (R = Real numbers) if and only if k does not equal blank3) Let C be a basis that contains vectors 2 (−2 + 1x + 1x^ + (−1) x3), (1 + (−2) x + 4x2 + 2x3), (−2 + 1x + 1x2 + (-1) 3), (−4 + 4x + 2 + 4x3) and P = C-B -1 Then : b₁ b2 = , b3 : = , ba = 3 = 4 4 -1 4 -24 -2 -2 2 ↓ NALN -1 -4 -2(c) Suppose ~ (A, B) is controllable. Suppose a change of basis is performed so that we obtain a new realization '~ (T-¹AT, T-¹B). Then ' is controllable.
- Let v1=( 0 1 -1 2)^T v2=(1 0 -2 1)^T Let V=Span{v1,v2} Find a basis for the orthogonal complement of V.Let B = {(1,1),(3,2)} and B′ = {(4,5),(2,4)} be two bases of R2. Let v be a vector with (v)B′ = (5, −6). E denotes a standard basis for R^2 (a) Based on the definition of coordinate vectors, compute v (in the standard basis) from the coordinate vector (v)B′ = (5, −6). (b) Find the transition matrix PB←B′ . (c) Use the transion matrix PB←B′ and (v)B′ = (5,−6) to find (v)B. (d) Based on the definition of coordinate vectors, compute v (in the standard basis) from the coordinate vector (v)B that you have obtained in part (c). Does it agree with your answer from part (a)? (e) Find the transition matrix PE←B′ . (f) Find the transition matrix PB←E. (g) Compute the product PB←E, PE←B′, and explain why the product equals to PB←B′ . I want anwers for part(d)(e)(f) and (g).20. (1) Give the coordinates of (4, 1, 5) in the basis (2) * Give the coordinates in the canonical basis of the point whose coordinates in the basis above are (3, 2,6) (3) Give the coordinates of (10, 3, 6) in the basis () 1 3