The 2-by-2 system is simple enough so that you can solve it by any means you know. However, the relations among N(A), C(A), solvable RHS, and all solutions, which should show up in the graphs, are the same for systems of any size or shape. Consider the following system of equations.
3x₁ - 2x₂ = b₁
6x₁ - 4x₂ = b₂

You will sketch two different graphs. You need relatively accurate graphs in order to answer part (f) which is the main point of this whole exercise.
(a) If we write the above system using matrix notations Ax = 5, what is the coefficient matrix A, what is 7, and what is b?
(b) What does it mean to say that the system is homogeneous? What is the definition of the nullspace of a given matrix? Describe the nullspace N(A) of this particular A in any way you can (pictures, equations, words, ...).
(c) What is the definition of the column space of a given matrix? Describe the column space C(A) of this particular A in any way you can.
(d) Choose three different right-hand vectors b ≠ 0 for any of which the system does NOT have a solution. On your first graph, sketch N(A), C(A), and these three b's.
(e) Choose two other different right-hand vectors b ≠ 0 for which the system DOES have a solution. Solve the system for each B's. How many solutions are there for each b? On your second graph, sketch N(A), C(A), these two b's, and all solutions corresponding to each b.
(f) Study your two graphs and notice any special relationship(s) among various objects. Complete the following sentences.
i. In order for At = 5 to have a solution, the right-hand vectors b (Hint: relate to C(A) and/or N(A)).
ii. In the case that Ac = b is solvable, all its solutions form C(A) (Hint: relate to C(A) and/or N(A)).

Airplane speeds are measured in three different ways: (1) indicated speed, (2) true speed, and (3) ground speed. the indicated airspeed is the airspeed given by an instrument called an airspeed indicator. a plane’s indicated airspeed is different from its true airspeed because the indicator is affected by temperature changes and different altitudes of air pressure. the true airspeed is the speed of the airplane relative to the wind. ground speed is the speed of the airplane relative to the ground. for example, a plane flying at a true airspeed of 150 knots into a headwind of 25 knots will have a ground speed of 125 knots. the problems below refer to static and dynamic pressure. static pressure is used when a body is in motion or at rest at a constant speed and direction. dynamic pressure is used when a body in motion changes speed or direction or both. a gauge compares these pressures, giving pilots an indicated airspeed. in problem #s 1 and 2, use the following information. the indicated airspeed s (in knots) of an airplane is given by an airspeed indicator that measures the difference p (in inches of mercury) between the static and dynamic pressures. the relationship between s and p can be modeled by s=136.4p√+4.5. 1. find the differential pressure when the indicated airspeed is 157 knots. 2. find the change in the differential pressure of an airplane that was traveling at 218 knots and slowed down to195 knots. in problem #s 3 and 4, use the following information. the true airspeed t (in knots) of an airplane can be modeled by t=(1+a50,000) ⋅ s, where a is the altitude (in feet) and s is the indicated airspeed (in knots). 3. write the equation for true airspeed t in terms of altitude and differential pressure p. 4. a plane is flying with a true airspeed of 280 knots at an altitude of 20,000 feet. estimate the differential pressure. explain why you think your estimate is correct.

What is the total area of the figure?

Given: ae ≅ ce ; de ≅ be prove: abcd is a parallelogram. we have that ab || dc. by a similar argument used to prove that △aeb ≅ △ced, we can show that △ ≅ △ceb by. so, ∠cad ≅ ∠ by cpctc. therefore, ad || bc by the converse of the theorem. since both pair of opposite sides are parallel, quadrilateral abcd is a parallelogram.