III Linear Transformations6. Label each of the following statements as true or false. If true,explain why briefly (no need to give a rigorous proof). If false, eithergive a counterexample or say how the statement should be modifiedto make it true. In all of the following, V and W are finite-dimensional vector spaces over a field F, and T : V → W.(a) If T(x + y) = T(x) +T(y) for all x, y E V, then T is linear.(b) T is one-to-one if and only if the only vector à such thatT(x) =0 is x = 0.=(c) If T is linear, then nullity(T) + rank(T) = dim(W).(d) If T is linear, then T maps linearly independent subsets of V tolinearly independent subsets of W.

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question #9
1. Let P(R) denote the vector space of polynomials with coefficients in R. Define a function T: P(R) → P(R) by T(p)(x) = p(x) — p(0). a. Show that T is linear. b. Is T one-to-one? Either show it is one-to-one or illustrate by example that it is not. c. Is T onto? Either show it is onto or illustrate by example that it is not.
Problem 3: Consider the set of polynomials {-1+3t – 21², 4 – 10t + 3t², 4 – 8t – 21²} . Let p¡(t) = – 1 + 3t – 21², p2(t) = 4 – 10t + 3ť², and p3(t) = 4 – 8t – 21². Let H = span {p,(t), P2(t), P3(t)}, which is a subset of P,.
Which types can be included in the function? that is formed by the basis of the g(x1, x2) function given with the truth table below.
In V, (R), where R is the field of real numbers, examine each of the following sets of vectors and find out which set of vectors is linearly independent (a) {(2, 1, 2), (8, 4, 8)} (b) {(- 1, 2, 1), (3, 0, – 1), (– 5, 4, 3)} (c) {(2,3, 5), (4, 9, 25)} (d) {(1, 2, – 1), (3, 1,5), (3, - 4, 7)}
Let F = Q(π3). Find a basis for F(π) over F.
11. Let the complex field C be a vector space over the real field R and the conjugation operator T on C defined by T(z) = z. 1 a. Find the matrix of T relative to the usual basis {1, i}. b. Find the matrix of T relative to the basis {1+i,1+2i}.
Let A E BL(H). Then A is called normal if A*A = AA*, uni- tary if A*A = I = AA*, that is, A* = A−¹, and self-adjoint if A* = A. Clearly, if A is unitary or self-adjoint, then A is normal. How- ever, a normal operator need be neither unitary nor self-adjoint. For example, let H = K² and for x = (x(1), x(2)) in H, A(x) = (x(1) − x(2), x(1) + x(2)). Then it can be easily seen that for x = H, A*(x) = (x(1) + x(2), −x(1) + x(2)), A*A(x) = 2(x(1), x(2)) = AA*(x). Hence A*A = AA*, but A*A ‡ I and A* # A. It is easy to see that if B is a normal operator and a C is a bounded operator such that C*C = I, then the operator A = CBC* is normal. Note that a bounded operator C may satisfy C*C = I without being a unitary operator. For example, let H = ² and for x = (x(1), x(2),...) in H, let C(x) = (0, x(1), x(2),...). 26 Normal, Unitary and Self-Adjoint Operators Then C*(x) = (x(2), x(3),…) for x € H, Hence Request explain Iin detail for better understan- 461 -ding C*C(x) = C*((0, x(1), x(2),...)) =…
Do fast
show that the inverse of arotation matrix is transpose,which is also arotation matrix, and it is generally not commulative
3 be W be vector a positive integer, V and field F, T: VV be linear transformation, and Spaces over f: V²->w be a multilinear alternating function. Let g: V" w be the function defined by Let n a V g (V., ..., Vn ) = + (T(V₁), T(V₂), ...,T (Un)) (V.... . Vn) EV" Prove the following statements. (a g is multilinear and alternating. (b) If V=F", W=F₁, and T=LA for some A € Mnxn (F), then g= (detA/.f
6. Suppose that V is a vector space over a field F and that v1, v2, ..., Vn € V. For each statement below, decide if the statement is true. If you believe it is true give a proof; if not, give an example that contradicts the statement. a. If v1, v2, ..., Vn are linearly independent then vị, v2, . .. , Vn is a basis of V. b. If vi E (v2, vV3, . .. , Vn) then V1, V2, . .. , Vn are linearly dependent. c. If vi E (v1, v2, V3, . . . , Vn) then v1, v2, . . . , Vn are linearly dependent. d. If v1, v2, ..., Vn are linearly independent then (v1, v2, ... , Vn) 7 (v2, ..., Vn)

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