数学偏微分方程代写 4143/6543代写 偏微分方程代写代写
2674143/6543 Partial Differential Equations 数学偏微分方程代写 All explanations and calculations must be shown in full to receive full credit. You may use the book but no electronic computation d...
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线性代数代写 Question 1. (40 Marks) This question consists of 20 multiple choice questions. Answer each question and put your answer in the table below.
This question consists of 20 multiple choice questions. Answer each question and put your answer in the table below. There is only one correct answer for each question.
Which one of the following is a subspace of R3 (under the usual operations in R3)?
(1) U = {(x, y, 1) : x = −y}.
(2) U = {(x, y, z) : x + y + z = 1}.
(3) U = {(x, 0, z) : x = z}.
(4) None of the above.
Which one of the following sets is linearly independent?
(1) {(1, 0, −1, 0),(0, 1, 0, 0),(1, 0, 1, 0),(0, 0, 0, 5)} in R4.
(2) {x + x2, 2x + x3, 2x2 − x3} in P3.
(3) {(0, 0, 0),(1, 1, 0).(1, 1, 1)} in R3.
(4) {(1, 1, 0),(1, 1, 1),(2, 4, −1),(3, 2, 2)} in R3.
Suppose U = span{x + x2, 1+2x + 2x2, 4, x3 + 4} (a subspace of P3). Then dim U =
(1) 1.
(2) 2.
(3) 3.
(4) 4.
Suppose U and W are subspaces of a vector space V . If dim V = 3, dim U = dim W = 2, and U ≠ W, then dim(U ∩ W) =
(1) 0.
(2) 1.
(3) 2.
(4) 3.
If U = {A ∈ M33 : AT = A}, then dim U =
(1) 3.
(2) 6.
(3) 2.
(4) 9.
Suppose T : V → W is an isomorphism. Which one of the following is TRUE?
(1) T(v) = 0W if and only if v = 0V , where 0V and 0W are the zero vectors in V and W, respectively.
(2) If V = span{v1, v2,…, vn}, then W = span{T(v1), T(v2),…,T(vn)}.
(3) dim V = 5 if and only if dim W = 5.
(4) All of the above.
If T : P3 → R is given by T(a + bx + cx2 + dx3) = d, then ker T =
(1) span{1, x, x2}.
(2) span{x, x2}.
(3) span{1, x}.
(4) {0}, where 0 is the zero polynomial in P3.
Define T : Pn → Pn via T(p(x)) = p(x) +xp’(x) for all p(x) ∈ Pn, where p’(x) is the derivative of p(x). Which one of the following is FALSE ?
(1) T is a linear transformation.
(2) ker T = {0}, where 0 is the zero polynomial in Pn.
(3) T is onto.
(4) T is not an isomorphism.
Let V = R3 (with the inner product being the usual dot product), and let U = span{(1, −1, 0),(−1, 0, 1)}, and v = (2, 1, 0). Then projU (v) =
(1) (1, 0, −1).
(2) (0, −1, 1).
(3) (2, 1, 0).
(4) None of the above.
Let P be an orthogonal matrix. Which one of the following is FALSE?
(1) det P = ±1.
(2) If det P = −1, then I + P has no inverse.
(3) PTP = I, where I is the identity matrix.
(4) None of the above.
Which one of the following is not necessarily true?
(1) Every hermitian matrix is normal.
(2) Every normal matrix is hermitian.
(3) Every unitary matrix is normal.
(4) All the eigenvalues of a hermitian matrix are real.
If T : P2 → P2 is given by T(p(x)) = p(x + 1), then det T =
(1) 1.
(2) 3.
(3) 4.
(4) 2.
Which one of the following is not necessarily true?
(1) If S and T are linear operators on V , where V is finite dimensional, then tr(ST) = tr(T S).
(2) If S and T are linear operators on V , where V is finite dimensional, then tr(S + T) = tr(S) + tr(T).
(3) If V is finite dimensional, the linear operator T on V has an inverse if and only if det T ≠ 0.
(4) If V is finite dimensional, the linear operator T on V has an inverse if and only if tr(T) ≠ 0.
Which one of the following is an inner product?
(1) V = R2, and h(a, b),(c, d)i = abcd.
(2) V = P3, and hp(x), q(x)i = p(1)q(1).
Let T be a linear operator on V , and let U and W be T-invariant subspaces of V . Which one of the following is not necessarily true?
(1) U + W is a T-invariant subspace of V .
(2) U ∩ W is a T-invariant subspace of V .
(3) U ∪W is a T-invariant subspace of V .
(4) T(U) is a T-invariant subspace of V .
Let V = R2 with inner product being usual dot product, and let u = v = (1, 1) and w = (−1, 0). Which one of the following is TRUE?
(1) {u, v, w} is an orthogonal set.
(2) ‖u + v + w‖2 = ‖u‖2 + ‖v‖2 + ‖w‖2.
(3) ‖u + w‖2 = ‖u‖2 + ‖w‖2.
(4) All of the above.
Let U = {A ∈ M22 : tr A = 0}. Prove that U is a subspace of M22.
Let T : V → W be a linear transformation.
(a) (5 Marks) If T is one to one and {v1, v2,…, vn} is independent in V , prove that {T(v1), T(v2),…,T(vn)} is independent in W.
(b) (5 Marks) If T is onto and V = span{v1, v2,…, vn}, prove that W = span{T(v1), T(v2),…,T(vn)}.
Define T : Mnn → Mnn via T(A) = A + AT .
(a) (5 Marks) Prove that T is a linear transformation.
(b) (5 Marks) Show that T is not an isomorphism.
Let v and w be vectors in an inner product space V .
(a) (5 Marks) Show that v is orthogonal to w if and only if ‖v + w‖ = ‖v − w‖.
(b) (5 Marks) Show that v + w and v − w are orthogonal if and only if ‖v‖ =‖w‖.
(10 Marks) Let V be an inner product space with inner product < , >, and let T : V → V be an isomorphism. Show that
<v, w>1 = <T(v), T(w)>
is an inner product on V .
(b) (5 Marks) Show that MB(T’ ) is the transpose of MB(T).
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