I will be talking from my experience with Robotics since that is where i used rotation matrixes the most. The rotation matrix when combined with a position vector (transformation matrix) can be used to describe the motion of the entire robot which is helpful when using a computers to control it but often times it is beyond time consuming to do it by hand. The transformation matrix for the entire robot is a multiplication of the transformation matrix of each joint which includes the rotation matrix. you can isolate the rotation matrix to a single joint by multiplying by the T inverse of the other joints

Rotation matrix for robotics simplifies a problem tremendously and gives a lot of information. ensuring a correct rotational matrix for robotics is essential to the proper function of the robot. Should the rotation matrix be wrong the computer will interpret the robotic arm as being a different geometry. this will cause problems with the calculation of the forward, inverse kinematics as well as the dynamics and for propagation which could cause the robot to crash or break

In practice orthogonal projection is taking a higher dimensional matrix and reducing the data into a lower dimensional matrix. by doing this it is easier to work with and most likely saves a tremendous amount of computational power which would be felt when it has to process a lot of information like the real time processing of video data. It is important to conserve information loss when reducing dimensionality

The inner product of functions is useful to find the orthogonality of functions. unlike before where we computed lengths when given specific points the product of functions determine the orthogonality when we are dealing with functions, in this case it is a sinusoidal pair of functions. when equation 3.37 is integrated and evaluated to 0 then the set of functions are othorgonal

Could linearly dependency be used as a safety check for a system. lets talk in reference to the problem below, since the west vector is linearly dependent on the the linear combination of the other 2 vectors could we not use it to ensure that the value of the independent vectors are correct? thus making the redundant vector valuable because it is redundant? (I used a simple example but couldn't this be useful in more complicated systems such as aerospace navigation?)

I am having trouble wrapping my head around the meaning of null space and application of the null space. Could someone explain the significance of it in a field such as robotics or other general applications? Thanks