Python Lab 5: Inner products, norms, and quadratic forms

Open 05_Inner_Products_Quadratic_Forms.ipynb and enter k=$k. Let u=\begin{bmatrix}1\\$k\\-1\end{bmatrix},\qquad v=\begin{bmatrix}$k\\1\\1\end{bmatrix}.

1. Compute u^Tv= inner product of u and v and \lVert u\rVert^2= squared norm of u .

   $dotUV=$dotExpected and $normUSquared=$normSquared

2. Compute the exact cosine of the angle between the vectors: \cos(\theta)= cosine of the angle between u and v .

   $cosTheta=$cosExpected

3. Is equality attained in the Cauchy--Schwarz inequality for these vectors?

   whether equality holds in Cauchy Schwarz Yes, because their dot product is positive. Yes, because their norms are equal. No, because the vectors are not scalar multiples. No, because Cauchy--Schwarz applies only to matrices. $csChoice.selectedIndex=3

4. Now let Q=\begin{bmatrix}2 & 1\\1 & $k+2\end{bmatrix},\qquad z=\begin{bmatrix}1\\-1\end{bmatrix}. Compute z^TQz= value of the quadratic form at z and \det(Q)= determinant of Q .

   $quadValue=$quadExpected and $detQInput=$detQ

5. What does the computation with np.linalg.eigvalsh show?

   classification of the quadratic form matrix Q is positive definite. Q is negative definite. Q is singular. Q is not symmetric. $pdChoice.selectedIndex=1