Rotating just the i^ vector (x-axis) i' = i cos φ is incomplete. Technically, you have to rotate every vector in the vector space. Anyway, a picture is worth a thousand words!



Given that T is a linear transformation. To show that it rotates every vector in 2-dimensional vector space R² counterclockwise through the angle θ, let v = (x, y) be a vector in R². Using polar coordinates (which you probably have learned in Critical Tutorial #3: Complex Numbers polar form), you can write v and T(v ) as



respectively, where r = √(x² + y²) is the length (a.k.a. norm) of vector v and α (alpha) is the angle from the positive x-axis counterclockwise to the vector v. Now, applying the linear transformation T to produces



So the projected coordinates (x',y') of the point (x,y) after rotation are:



From the linear transformation above, R(θ) is known as the counterclockwise Rotation matrix:



If the direction of vector rotation is clockwise, then the Rotation matrix becomes:

Hi maximR,

You have seen how linear transformation works. Here is one real application of linear transformation. A robot can drive a certain distance or turn about a certain angle in its local coordinate system. For many applications such as Unmanned ground vehicle (UGV), Unmanned aerial vehicle (UAV), and Unmanned underwater vehicles (UUV), however, it is important to first establish a map (in an unknown environment) or to plan a path (in a known environment). These path points are usually specified in global or world coordinates.

Translating local robot coordinates to global world coordinates is a 2D transformation that requires a translation and a rotation, in order to match the two coordinate systems as shown in the Figure below.



Assume the Robot has the global position [Rx, Ry] and has global orientation ψ. It senses an Obstacle at local coordinates [Ox', Oy']. Then the global coordinates [Ox, Oy] can be calculated as follows:



For example, the marked position at the above Figure has local coordinates [0, 3]. The robot’s position is [5, 3] and its counterclockwise orientation is 30°. The global object position is therefore:



For computation purpose (perhaps kingkingyyk is interested ), the coordinate transformations can be greatly simplified by using homogeneous coordinates, where an arbitrary 2-D transformation sequences can be summarized in a single 3×3 matrix.

Frankly speaking, Not Enough! There are several textbooks of unquestionable merit for Physics, but I personally keep these two with me. Try finding them at Sunway Library.

Physics for Scientists and Engineers with Modern Physics, 9e / Serway & Jewett



University Physics with Modern Physics, 13e / Young & Freedman

Hi maximR

A homicide victim is discovered and a pathologist from the forensics laboratory is summoned to the death scene to estimate the time of death. The strategy is to find an expression T(t) for the body’s temperature at time t, taking into account the fact that, after death the body will cool by radiating heat energy into the room. T(t) can be used to estimate the last time at which the victim was alive and had a “normal” body temperature. This last time was the time of death.

To find the cadaver temperature T(t), some information is needed. First, the pathologist finds that the body is located in a room that is kept at a constant 20°C. For some time after death, the body will lose heat into the cooler room. Assume, for want of better information, that the victim’s temperature was 37°C at the time of death. By Newton’s law of cooling, heat energy is transferred from the body into the room at a rate proportional to the temperature difference between the room and the body. If T(t) is the body’s temperature at time t, then Newton’s law says that, for some constant of proportionality k,

.

This ordinary differential equation (ODE) is separable, since algebra allows us to rewrite the ODE so that each of two variables occurs on a different side of the equation:



Integrate both sides to obtain (see Critical Tutorial #6)



To solve for T, take the exponential of both sides of this equation to get



Now the constants k and B must be determined. Since there are two constants, we will need two pieces of information. Suppose the pathologist arrived at 9:40 p.m. and immediately measured the body temperature, obtaining 34.7°C. It is convenient to let 9:40 p.m. be time zero in carrying out measurements. Then



To determine k, we need another measurement. The pathologist takes the body temperature again at 11:00 p.m. and finds it to be 31.8°C. Since 11:00 p.m. is 80 minutes after 9:40 p.m., this means that



The temperature function is now completely known as





The time of death was the last time at which the body temperature was 37°C (just before it began to cool). Solve for the time t at which



According to this model, the time of death was approximately −53 minutes. It means death occurred approximately 53 minutes before (because of the negative sign) the first measurement at 9:40 p.m., which was chosen as time zero in this model. This puts the murder at about 8:47 p.m.

This is why I insisted you and RED-HAIR-SHANKS to learn differential equations seriously and EARLY. You'll probably ask, Why differential equations? Why are they so important to deserve your attention?

Well, differential equations arise naturally as models in areas of science, engineering, economics, and lots of other subjects. Physical systems, biological systems, and economic systems; all these are marked by change, or dynamics. Differential equations model real-world systems by describing how they change. The unknown function y(t) could be the current in an electrical circuit, the concentration of a chemical undergoing reaction, the population of an animal species in an ecosystem, or the demand for a commodity in a micro-economy. Differential equations represent laws that dictate change, and the unknown y(t), for which we solve, describes exactly how the changes occur.

You don't have touch on Partial Differential Equation (PDE) at the moment until you learn Partial Differentiation later. Just focus on First-order ODEs and familiarize with solutions of ODEs. You'll need to learn to identify certain types of 1st-order ODEs such as Separable, Linear, Homogeneous, Bernoulli, and Riccati ODEs because different types have different techniques to solve them. After mastering Separation of Variables, the next one to learn is the integrating factor:

.

It may look complicated at first glance, but don't judge the formula by its Integral operator ∫.
Good question! I purposely left out some explanations earlier, and you're up to my expectation. Here is the explanation:

This is just a prediction (but a good one), because an educated guess (37°C) was made of the body’s temperature before death. It is also impossible to keep the room at exactly 20°C (according to Fluid dynamics and Chaos theory). However, the model is robust in the sense that small changes in the body’s normal temperature and in the constant temperature of the room yield small changes in the estimated time of death. This can be verified by trying a slightly different normal temperature for the body, say 37.4°C, to see how much this changes the estimated time of death.

P.S.: A model is said to be ROBUST if it can make “good” prediction under the worst-case conditions on the “unknown unknowns” in the system dynamics. Robustness is of crucial importance in system modeling, because real dynamic and engineering systems are vulnerable to external disturbance and measurement noise, and there are always discrepancies between mathematical models used for calculation and the actual system in the real world.

Do I need to know this on my level?

is this what you're talking about?