Will these measurements (f(x) and f'(x) at two distinct points) always yield a linearly independent matrix? I imagine it does because this method is so widely used, but I can't think of an intuitive reason as to why this would always be the case.
killawhale2
Because the four equations that are given are independent of each other. Think of it this way, if I tell you the value of a polynomial at a certain point, does this tell you anything about its derivative at the same point? Likewise, what would knowing the derivative of a polynomial at a given point tell you about the actual values at the said point?
Will these measurements (f(x) and f'(x) at two distinct points) always yield a linearly independent matrix? I imagine it does because this method is so widely used, but I can't think of an intuitive reason as to why this would always be the case.
Because the four equations that are given are independent of each other. Think of it this way, if I tell you the value of a polynomial at a certain point, does this tell you anything about its derivative at the same point? Likewise, what would knowing the derivative of a polynomial at a given point tell you about the actual values at the said point?