Gif a is a real n×n matrix, then we have seen that < v, w > a: = v t ∗a∗ w defines a bilinear pairing on r n which is symmetric iff a is a symmetric matrix. if a is symmetric, we call a positive definite iff < v, v > a > 0 for all 0 6= v ∈ r n (in other words, < −,− > a is an inner product on r n). a) [1 pt] suppose b is the matrix given by b = 1 1 1 2 and x = [x1 x2] t ∈ r 2 . find an expression for x t ∗ b ∗ x in terms of x1 and x2. b) [1 pt] using your work in part a), show that the matrix b above is positive definite.

What is the exact volume of the cylinder? 16in going into the middle of the top of the cylinder and 40in in height a) 25,600(pi)3 b)10,240(pi)3 c)1280(pi)3 d)640(pi)3

You have learned about the six trigonometric functions, their definitions, how to use them, and how to represent them graphically. the sine, cosine, and tangent trigonometric functions can be paired with their reciprocal functions, cosecant, secant, and cotangent, respectively. think about how each function is related to its reciprocal function.how are the graphs of the reciprocal functions related to their corresponding original functions? what happens to the graphs of the reciprocal functions as x approaches the zeros of the original functions? describe how you would teach friends with different learning styles (visual-spatial, aural-auditory, verbal-linguistic, physical-bodily-kinesthetic, logical-mathematical, social-interpersonal, and solitary-intrapersonal) how to graph the reciprocal functions

In the given triangle, ∠aed ∼ ∠ abc, ad = 6.9, ae = 7.2, de = 5.2, and bc = 10.2. find the measure of bd and ce. round your answer to the nearest tenth.