Linear Algebra

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Section 5.5 Orthonormal Sets: Problem 1 (1 point) Find the missing coordinates such that the three vectors form an orthonormal basis for IR" : -0.6 -0.8 -0.6Section 2.1 Determinant: Problem 11 (1 point) Given the matrix a A = a 5 5 a find all values of a that make the A|= 0. Enter the values of a as a comma-separated list: 3.51,-1.63Section 5.5 Orthonormal Sets: Problem 10 (1 point) 5 5 -4 4 -3 Let y = VI= A 2 .4 48 Compute the distance d from y to the subspace of IR spanned by v, and v2. dSection 5.5 Orthonormal Sets: Problem 11 (1 point) Are the following statements true or false? 1. If y = %1 + %2, where % is in a subspace W and %2 is in W, then % must be the orthogonal projection of y onto W. 2 2. The best approximation to y by elements of a subspace W is given by the vector y - projw (y)- 2 3. Let 1], 112, . .., Un be an orthonormal basis of a vector space V. In the Orthogonal Decomposition Theorem, each term p = (b, uj)uj +... + (b, Un)W, is itself an orthogonal projection of b onto a subspace of V. 4. If an n x p matrix U has orthonormal columns, then UU? x = x for all x in IR". v 5. If W is a subspace of IR" and if v is in both W and WI, then v must be the zero vector. True N False to get credit for this problem all answers must be correct.Section 5.5 Orthonormal Sets: Problem 12 (1 point) [0.5 -0.57 0.5 -0.5 Let A = 0.5 0.5 Note that the columns of A are orthonormal (why?). 0.5 0.5 (a) Solve the least squares problem Ax = b where b = (b) Find the projection matrix P that projects vectors in R* onto R(A) P = (c) Compute Ax and Pb Ax = Pb =Section 5.5 Orthonormal Sets: Problem 2 (1 point) Let 0.5 -0.5 0.5 0.5 0.5 -0.5 VI = 0.5 > US = -0.5 0.5 0.5 0.5 0.5 Find a vector w, in IR* such that the vectors v1, 12. us, and us are orthonormal. JA =Section 2.2 Properties of Determinants: Problem 6 (1 point) If a 4 x 4 matrix A with rows v1, v2, v3, and v4 has determinant det A = -9, 501 + 502 3v1 + 6v2 then det 0 U3 VASection 5.5 Orthonormal Sets: Problem 3 (1 point) -5 Use Theorem 5.5.2 to write the vector v = 9 as linear combination of 3/ v 10 3/v18 3/V342 1 = 1/ V19 102 = 0/V18 and U; = -18/V342 3/v19 -3/V18 3/1342 Note that uj, uz and us are orthonormal. V uit Use Parseval's formula to compute | |v| |?.Section 3.1 Vector Spaces: Problem 3 (1 point) Let V be the set of vectors in R. with the following definition of addition and scalar multiplication: Addition: [2 + y/2 Scalar Multiplication: o O Determine which of the Vector Space Axioms are satisfied. Al. xOy = y @ x for any x and y in V YES v A2. (x (y) Oz = x@ (y @z) for any x, y and z in V YES v A3. There exists an element Ov in V such that x D Ov = x for each x e V NO A4. For each x = / f(x)g(x)dx in the vector space CO[0, 1] to find the orthogonal projection of f(x) = 512 - 4 onto the subspace V spanned by g(x) = r - - and h(x) = 1. projv (f) =}Section 5.5 Orthonormal Sets: Problem 7 (1 point) Are the following statements true or false? V 1.uv- vu=0. 2. If u and v are nonzero vectors and |lull* + lly| = |lu + v|, then u and v are orthogonal. V 3. If x is orthogonal to every vector in a subspace W, then X is in WI. 4. For an m x in matrix A, vectors in the null space of A are orthogonal to vectors in the row space of A. 5. For any scalar c, Ilevll = cllull. True N False to get credit for this problem all answers must be correct.Section 5.5 Orthonormal Sets: Problem 8 (1 point) Are the following statements true or false? 1. If y is in a subspace I, then the orthogonal projection of y onto W is y itself. V 2. If % is orthogonal to uj and uz and if W = span(U1, u2), then % must be in W L 3. For each y and each subspace W, the vector y - projw(y) is orthogonal to W. 4. If the columns of an n x p matrix U are orthonormal, then DU y is the orthogonal projection of y onto the column space of U 5. The orthogonal projection P of y onto a subspace W can sometimes depend on the orthogonal basis for W used to compute p True N False to get credit for this problem all answers must be correct. Preview My Answers Submit AnswersSection 5.5 Orthonormal Sets: Problem 9 (1 point) Suppose v1, V2, vs is an orthogonal set of vectors in 1R3. Let w be a vector in span (v1, V2, vs ) such that (VI, VI) = 45, (v2, V2) = 9, (Vs, V3) = 9, (w, vi) = -225, (w, v2) = -18, (w, Vs) = 9, then w VI V2 +