Justify your answers.Post by Professor Hovasapian on March 23, 2013 8.9.Determine whether the following sets are subspaces of R3 under the operations of addition and scalar multiplication defined on R3. Prove that diagonal matrices are symmetric matrices. Prove that tr(aA + bB) = a tr(A) + b tr(B) for any A, B ∈ Mn×n (F ). Prove that A + At is symmetric for any square matrix A. Prove that (At )t = A for each A ∈ Mm×n (F ). In addition, if the matrix is square, compute its trace. Determine the transpose of each of the matrices that follow. How many suites were sold during the June sale? Let S =. If the inventory at the end of June is described by the matrix ⎛ ⎞ 5 3 1 2 A = ⎝6 2 1 5⎠, 1 0 3 3 7.8.9.interpret 2M − A. Assuming that none of the present stock is sold until the additional furniture arrives, verify that the inventory on hand after the order is filled is described by the matrix 2M. To prepare for its June sale, the store decided to double its inventory on each of the items listed in the preceding table. Early Mediter- American Spanish ranean Danish Living room suites 4 2 1 3 Bedroom suites 5 1 1 4 Dining room suites 3 1 2 6 Record these data as a 3 × 4 matrix M. At the end of May, a furniture store had the following inventory. 1 Vector Spaces Fall Spring Summer Brook trout 9 1 4 Rainbow trout 3 0 0 Brown trout 1 1 0 Record the upstream and downstream crossings in two 3 × 3 matrices, and verify that the sum of these matrices gives the total number of crossings (both upstream and downstream) categorized by trout species and season. Upstream Crossings Fall Spring Summer Brook trout 8 3 1 Rainbow trout 3 0 0 Brown trout 3 0 0 14 Downstream Crossings Chap. Wildlife Management, 25, 221-242) reports the following number of trout having crossed beaver dams in Sagehen Creek. Richard Gard (“Effects of Beaver on Trout in Sagehen Creek, Cali- fornia,” J. (k) Two functions in F(S, F ) are equal if and only if they have the same value at each element of S. 1.2 Vector Spaces 13 (j) A nonzero scalar of F may be considered to be a polynomial in P(F ) having degree zero. If f is a polynomial of degree n and c is a nonzero scalar, then cf is a polynomial of degree n. If f and g are polynomials of degree n, then f + g is a polynomial of degree n. In P(F ), only polynomials of the same degree may be added. An m × n matrix has m columns and n rows. A vector in Fn may be regarded as a matrix in Mn×1 (F ). In any vector space, ax = ay implies that x = y. In any vector space, ax = bx implies that a = b. A vector space may have more than one zero vector. (a)(b)(c)(d)(e)(f )(g)(h)(i)Every vector space contains a zero vector. 1.Label the following statements as true or false. Prove that the diagonals of a parallelogram bisect each other. 6.Show that the midpoint of the line segment joining the points (a, b) and (c, d) is ((a + c)/2, (b + d)/2). 5.Prove that if the vector x emanates from the origin of the Euclidean plane and terminates at the point with coordinates (a1, a2 ), then the vector tx that emanates from the origin terminates at the point with coordinates (ta1, ta2 ). 1.EXERCISES Determine whether the vectors emanating from the origin and termi- nating at the following pairs of points are parallel.
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