![]() (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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