2201NSC Introduction to Mathematical Modelling

Introduction to Mathematical Modelling

We’ve developed a lot of the basic ideas of Linear Algebra in the context of solving systems of linear algebraic equations. Our vectors were lists of n values, x = (x₁; x₂; : : : ; x_n), representing possible values of the n variables found in our system of equations. We represented the equations in the form Ax = b, and set about looking for possible solutions. More recently, we’ve considered cases where the matrix A represents (possibly huge) sets of data, which we want to analyse in order to make predictions (say, about the best song to recommend to a user on a music platform).

An alternative use of linear algebra, which we’ve also considered, is when the values in x represent coe cients for a set of basis functions. We then extend all the ideas we’ve developed, like linear dependence, orthogonality, and so on, to help solve mathematically similar problems. Here we use inner products to measure how ‘similar’ two vectors are.

points in the direction closest to u = (1; 2; 1; 2; 1; 2)^T , measured using the standard inner product (the dot product)?

Q2 Which of the six vectors v₁ to v₆ above points in the direction closest to u = (1; 2; 1; 2; 1; 2)^T , measured using the inner product

hx; yi = 2x₁y₁ + x₂y₂ + 2x₃y₃ + 2x₄y₄ + x₅y₅ + 2x₆y₆   ?

Q3 Which of Users 1-5 has taste in music most similar to User 6, measured using the standard inner product?