Solution

Solution

Solution

Let us first find an orthogonal basis for W

Now, to avoid fractions in our computation, let us consider 3w2, instead of

w2. Note that the scaling does not change the orthogonality.

We have

Now it suffices to normalize the vectors w1, 3w2 to obtain an orthonormal

basis. Therefore, the set

Problem

Solution

Given 2x2 +12xy +20y2 is positive definite ?

Solution

The matrix

Is positive definite – its determinant is 4 and its trace is 22 so its eigenvalues are positive

Problem