G 4. (5 points) If A is an m × n matrix, prove that the null space of AT and the column space of A are orthogonal complements in R m. For example, show that N(AT ) = (C(A))⊥ . (Note that C(A) = N(AT ) ⊥ is also true). Note: Because of Theorem 4.8.6 (c) See Text Page 253, that is (W⊥) ⊥ = W, once we have N(A) = (R(A))⊥ , then R(A) = (N(A))⊥ will also follows. Similarly, once we have N(AT ) = (C(A))⊥ , then C(A) = N(AT ) ⊥ will also follows. Note: The results obtained in Problem 3 and 4 are also refered to as Fundamental Theorem of Linear Algebra, this shows its important in the subject

An airplane consumes fuel at a constant rate while flying through clear skies, and it consumes fuel at a rate of 64 gallons per minute while flying through rain clouds. let c represent the number of minutes the plane can fly through clear skies and r represent the number of minutes the plane can fly through rain clouds without consuming all of its fuel. 56c+64r < 900056c+64r< 9000 according to the inequality, at what rate does the airplane consume fuel while flying through clear skies, and how much fuel does it have before takeoff? the airplane consumes fuel at a rate of gallons per minute while flying through clear skies, and it has gallons of fuel before takeoff. does the airplane have enough fuel to fly for 60 minutes through clear skies and 90 minutes through rain clouds?

The number of chocolate chips in an 18-ounce bag of chocolate chip cookies is approximately normally distributed with a mean of 1252 chips and standard deviation 129 chips. (a) what is the probability that a randomly selected bag contains between 1100 and 1500 chocolate chips, inclusive? (b) what is the probability that a randomly selected bag contains fewer than 1125 chocolate chips? (c) what proportion of bags contains more than 1225 chocolate chips? (d) what is the percentile rank of a bag that contains 1425 chocolate chips?

Is appreciated! graph the functions and approximate an x-value in which the exponential function surpasses the polynomial function. f(x) = 4^xg(x) = 4x^2options: x = -1x = 0x = 1x = 2