Math 54 - Practice Final1. Let V be the vector space of real 3 × 3 matrices. Let W be the set of matrices A ∈ V such thatAT= −A. Is W is a subspace of V ? If so, find a basis for W .2. Mark each of the follow statements TRUE, FALSE, or NONSENSE, where NONSENSE indicatesthat one or more vocabulary terms have been misused in the statement. No justification is required.(a) Every solution to the heat equationut= α2uxx,u(0, t) = u(1, t) = 0can be written in the form u(x, t) = T (t)X(x).(b) If P (t) is an n × n matrix whose columns are solutions to the system of differential equations~x(t) = A~x(t), then P (t) is a fundamental matrix of the system.(c) If the columns of an m × n matrix A are linearly independent, then A spans Rm.(d) If A, B, and C are symmetric n × n matrices, then AB + C is symmetric.(e) If A and B are matrices such that AB = In, then A and B are both invertible.(f) If A is a matrix, thenRow(A) = Row(ATA),where Row denotes the row space.(g) If W is any subspace of an inner product space V , then W⊥∩ W = {~0}, where ∩ indicatesintersection.3. Find the Fourier Series expansion of the functionf(x) =0 for −1 ≤ x ≤ 01 for 0 < x ≤ 1on the interval [−1, 1]. As usual, write your answer in the formf(x) ∼a02+∞Xn=1ancos(nπx) +∞Xm=1bmsin(mπx).4. Consider the following partial differential equation.uxx+ 2ux= ut,u(0, t) = u(π, t) = 0,u(x, 0) = f (x),where u(x, t) is defined for t > 0 and 0 < x < π.(a) What would a function of the form u(x, t) = X(x)T (t) need to satisfy in order for it to be asolution to the PDE above? State your answer in terms of ODEs in functions X(x) and T (t), aswell as any boundary conditions the ODEs would satisfy.(b) Find a solution to the PDE that satisfies the initial conditionu(x, 0) = e−xsin x − e−xsin 2x.5. Let T : V → W be a one-to-one and onto linear transformation. Show that dim V = dim W .6. Consider the system of linear differential equations~x0(t) =−1 c1 −1x(t).1(a) For which values of the constant c does every solution ~x(t) have the property thatlimt→∞~x(t) =~0?(b) For which values of c does there exist a non-zero solution such that x1(t) = 0 for infinitely manyvalues of t, where ~x(t) = (x1(t), x2(t))T.7. Let A be a matrix whose null space is {0}. Explain carefully why each of the following statements istrue:(a) The rank of A equals the number of columns of A.(b) The rows of A are linearly independent if and only if A is a square matrix.(c) The product ATA of the transpose of A and A is an invertible matrix.8. Let V be the vector space of all continuous functions on the real line. Consider the inner producthf, gi =Z10f(x)g(g) dxon V . Find a non-zero function that is orthogonal to the constant function 1 and to the function xand
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