Geometry and Determinants
Math 272  Prof. Leise

Email completed labs to tleise@amherst.edu by 4pm Tuesday.

To take a determinant in Mathematica, use the following commands:

A={{1,2,0,4},{5,0,7,8},{9,10,11,12},{13,14,15,15}};A//MatrixForm
Det[A]

Important Theorem: A square matrix A is invertible if and only if det(A) is not zero.

Equation of line through two points

We want to find an equation ax+by+c=0 for the line through two given points (x1,y1) and (x2,y2).  Plugging each point into the equation gives us a total of 3 equations in 3 unknowns a,b,c:

ax+by+c=0
ax1+by1+c=0
ax2+by2+c=0

We want a,b,c not all equal to zero.  That is, we want a NONTRIVIAL solution to the HOMOGENEOUS system.

Exercise 1.  Suppose we want the line to pass through the points (1,2) and (3,5).  Rewrite the system as a matrix equation Ax=0 (treat x and y as coefficients of a and b; the matrix A should involve x, y, and numbers, but not a, b, or c). What should the determinant of A equal, given that A is not invertible?

Exercise 2.  Compute the determinant of A and set it equal to the value you gave in Exercise 1. This gives the equation of the line through (1,2) and (3,5).

Parabola through three noncollinear points

Let's do a similar procedure to find the equation ax^2+bx+cy+d=0 for a parabola through noncollinear points (x1,y1), (x2,y2), and (x3,y3).

Exercise 3. Suppose we want the parabola to pass through (-1,6), (0,1), and (1,0).  Use these three points to generate three more equations.  Form a matrix equation of the original parabola equation in x and y plus the three equations from plugging in the points.

Exercise 4. We want a nontrivial solution to this homogeneous system Ax=0, so again A is not invertible.  Simplify det A=0 to obtain the equation of the parabola through the three given points.

Circle determined by three noncollinear points

Exercise 5.  Use a similar procedure to find an equation a(x^2+y^2)+bx+cy+d=0 for the circle containing the points (-1,5), (2,2), and (-4,2).  What is the center and radius of this circle?  (Recall that a circle with radius R and center at (x0,y0) is described by (x-x0)^2+(y-y0)^2=R^2.)

Test for linear independence

An nxn matrix A is invertible if and only if detA is not zero.  We also know that a matrix is invertible if and only if Ax=0 has only the trivial solution. Another interpretation of Ax=0 having only the trivial solution is that the only linear combination of the columns of A that equals zero is to multiply each column by 0: 

x1*Column1 + x2*Column2 + x3*Column3 = ZeroVector implies x1=x2=x3=0.

We then say that the columns of A are linearly independent (meaning that no column is a linear combination of the other columns).  

We can test whether the columns of A are linearly independent by checking whether the determinant of A is zero.  If detA=0, the columns are not linearly independent; otherwise they are linearly independent. We can extend this to a general set of vectors by putting them as the columns of a new matrix A and taking the determinant.

Exercise 6.  Is the set {v1,v2,v3,v4} of vectors v1=(({{({{1}, {2}})}, {({{4}, {3}})}})), v2=(({{({{-1}, {0}})}, {({{2}, {1}})}})), v3=(({{({{0}, {1}})}, {({{2}, {3}})}})), v4=(({{({{1}, {-1}})}, {({{-1}, {3}})}})) linearly independent? Test using a determinant.

Exercise 7.  Is the set {v1,v2,v3,v4} of vectors v1=(({{({{1}, {2}})}, {({{4}, {3}})}})), v2=(({{({{-1}, {0}})}, {({{2}, {1}})}})), v3=(({{({{1}, {1}})}, {({{5}, {7}})}})), v4=(({{({{1}, {-1}})}, {({{-1}, {3}})}})) linearly independent? Test using a determinant.

Wronskian determinant

We can also test whether a set of functions is linearly independent, in the sense that one function cannot be written as a linear combination of the others in the set.  This idea is important, for instance, in differential equations, where you want to write the general solution as a linear combination of linearly independent functions.
    
We can test for independence by evaluating the determinant composed of the functions and their derivatives.  For example, to test the linear independence of {f(x), g(x), h(x)}, check whether the determinant of the Wronskian matrix (({{f[x], g[x], h[x]}, {f'[x], g'[x], h'[x]}, {f''[x], g''[x], h''[x]}})) is zero.  Identically zero implies dependent, not zero for at least one value of x implies independent.

To take derivatives, you can use the following commands if you like:

f[x_]:=x^3; (* defines a function f(x) *)
D[f[x],x]       (* first derivative of f(x) *)
D[f[x],x,x ](* second derivative of f(x) *)

Exercise 8.  Is the set of functions {Exp[x], Exp[2x], Exp[3x]} linearly independent?

Exercise 9.  Is the set of functions {Exp[x], Exp[-x], Sinh[x]} linearly independent?

Exercise 10.  Is the set of functions {1, x, x^2} linearly independent?