Tutor profile: Saksham S.
Subject: Physics (Newtonian Mechanics)
What are 'fictitious' forces?
Fictitious forces, also called Psuedo forces are those that arise when we are observing any body from a frame of reference which itself is accelerating! Now, all definitions of forces, acceleration, velocity etc are with respect to a frame of reference. Imagine you are in an airplane which is taking off at the runway and you're holding a bottle of water. The water surfaces gets slanted but how is that when you are not applying any force to it? To explain the water behavior, we need to conjure up a new force which we say is acting on the water but only because we happen to be in an accelerating frame ( the airplane). This force we need to input to our equations of motion to explain the observed behavior. Once the airplane stops/reaches constant speed, the water surface becomes horizontal and the force vanishes too!
Subject: Linear Algebra
What is the meaning of a null space of a matrix?
Every matrix multiplication operation is something that maps a vector from one space to another space. Lets say we have a 3x3 matrix A which after gaussian elimination becomes 2x3 which we apply to a vector X (3x1) to give an output Y(2x1). THis means that it transformed a 3-d vector to a 2-d vector or in other words it squeezed the 3-Ds of the space in which X was to 2-Ds in which Y lies. IMagine if we are to smash a paper box into a flat square, there are so many points along the direction in which we smashed it which all fall onto one point on the square. The line of points which happen to fall on the origin ( let it be the center of the smashed square) are the null space of the matrix.
How to really 'understand' Fourier series?
Imagine if we could represent an arbitrary function as being built up of standard, known and easy to work with functions, wouldn't it make the math a lot easier? This is what Fourier series helps us with. The magic is to break any function into sines and cosines to make the subsequent math easier. How does it do that? Remember, we can write a vector in 2-d space as a sum of i-hat and j-hat vectors that represent x and y axes: ex: 5i + 6j. in 3-D: 5i+6j+8k (say). But lets not stop there and keep expanding the dimensions beyond the boundaries of imagination to infinity: 5i+6j+9k+4l+7h...so on. Each of these i,j,k... are equivalent to the sine and cosine functions in a fouries series. Why choose arbitrary sine and cosine functions? When were were writing a vector a sum of i,j,k.. each of these i,j,k were orthogonal (90 degree) to each other right? This means their dot product is zero. It turns out, this is also a special property of sines and cosines: their 'dot product' is also zero! How to prove this? Just like a vector in 3-D space is a set of 3 numbers, we can imagine infinitely many y values from the sine and cosine function to represent an 'infinite'-D space. Just like in case of dot product where we multiply the i's, j's, k's together and then sum them to get the value, we can do the same to numbers of sine and cosine and add them. Since they are infinitely close to each other on the number line, this addition is equal to using integration.
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