1.2 Background: Linear Algebra This project will require very basic linear algebra - no more than matrix multiplication and the tools to calculate the required work will be provided so you can focus on the math. Consider the following matrix equation: ay) = (3 31) (:)'(_11)=(g::r:ii) If we dene A := (3 31), :E' := (Z), and I: := (11) then we can write this expression as ALE 3. Now observe that this expression is functionally equivalent to the following: Thus there are two ways to write the function f (9:, y) - as a either a matrix equation, or as a typical multivariate function. Symbolab is a great tool if you are unfamiliar with matrix multiplication. Click here to see the matrix multiplication above done in symbolab. Also, while we can't graph f (:13, y) fully, we have some tools to analyze this function: link. The motivation behind this project comes from an important topic in linear algebra - solving systems of equations. For example, we might be interested in know for which values of x and y the following is true: 1 1 3$+y 2xy This is equivalent to asking when f (:13, y) = (0,0) or equivalently A53" 3: 6. 1.3 Background: The Gradient For a multivariate function h from R\" > R, we can construct the gradient of h, denoted Vh. When n = 2 (the only case we consider) the gradient of h : R2 > R is: a a 83' By For example, the gradient of m(x, y) = ln(:c3 + 'y) is 2 Vm= 3a: 1 1 x3+y x3+y Vh = Vh(x, y) = R. (For example, the length of the vector (2,5) is H (2, 5) II = V22 + 52 = \\/29. Don't let the fact that f (ac, :9) depends on :1: and 3;). What condition is necessary for a vector ((1,1)) to have zero length? Now compute VII f (m, y)\"2 = VL(:1:, y). Note that your answer should be a function VL : R2 > R2. Now compute AT(AzE' 5') Here AT is the transpose of A, so AT = (i) 31). Again, if you are unfamiliar with matrix multiplication, the link above to symbolab is a great tool to carry out this calculation. Your answer should be a 2 X 1 matrix (vector). Compare the results from the previous two steps. Use this to guess the general mama formula for the gradient of the function IIAJE' 3H2. That is, complete the following equation: _' VIIAzi' I)\"2 =