MATH 218 Final, Spring 2004

Math.218. Final (120 minutes). June2004. R. Barbara. 1 Answer by true or false: _Every subset of a linearly independent set is a linearly independent set. _ The vectors (1, 0, 1), (1, 1, 0), (3, 2, 0) form a basis for R*. _The column vectors of a 7x4 matrix must be linearly dependent. _If span(S) = span(T), then, S = T. _lf {a, b, c} is a linearly independent set of nonzero vectors, then, each is a linear combination of the others. _If u,v are vectors of a inner product space, then, {ee =| if and only if u—v is orthogonal to u tv. 2 If we modify the scalar multiplication in R? by setting k(x,y) = (2kx, 2ky), does R? remain a vector space ? Pp ry g y 13 3 Is W={(x yz) eR*®, y=x+z+1} a sub-vector space of R* ? 4 Prove or disprove that x = (3, 1, 5) isa linear combination of u = (0, -2, 2) and v=(1, 3,-1). 5 Does the solution set of the linear system x+y+z=0, x—2y+z=1 forma sub-vector space of R° ? 6 Determine whether the vectors u = (2, -1,3), v= (4, 1,2), w=(0,3,-4) forma basis for R°. 7 Are the vectors a=(0,0, 1,2), b=(5, 1,2, 0), and c=(l, 0, 0, 3) linearly independent in R’? ; ; 2 0 : ; -1 1 fl 1 0 0 0 0 8 Find the coordinate vector of relative to the basis A, = > A= »A;= » A= . -1 3 0 0 [9 0 1 0 0 J 9 Find a basis for the nullspace of A, andthe rank of A, where A= 11 Consider the weighted inner product on R* defined by <u, v>=3.u,v,; +5.u5V, . eFind the norm of u = (-8, 15). eFind the distance between u and v = (17, —3). Find the cosine of the angle between u and v. 12 Let T be the linear transformation: R‘—> R? defined by T(x, y, z, t)= (40x -20z + 5t, x+y+z+t, 6t). Find a basis for the Kernel of T. Subsidiary question: Is it possible to modify the numerical values above in order to obtain rank(T) = 4 ? 3-1 0 13 Let n bea positive integer, andletA=|-1 2 0 -1 3 A 4 3 14 Let t denote a real parameter. Consider the 3x3 matrix A= A —24 A VA . Find all the values of 1 aA A for which matrix A is orthogonal.(a complete proof is needed). 15 Prove or disprove: We can find a 1x3 matrix M such that, for infinitely many 3x1 vectors v, Mv is a prime number. 16 Let p and q be real numbers and suppose that the equation x*+ p-x A 4x4 q=0 has at least one real zeros. Prove the existence of a 4x4 matrix A satisfying A’ +A = p.A?+g.l p Gl, 17 The following property is well-known: If A is a mxn matrix with rank n, then, A’ A is invertible. Does the result remain true if we allow complex entries ? (104+4+4+44+44+54545474+4+94+8+9+1044+44+4)