Recall that the image of a matrix \(A \in \mathbb{C}^{m \times n}\) is the set \(\operatorname{Im} A = \{Ax : x \in \mathbb{C}^n \}\). The image is also known as the range or column space of \(A\). The kernel of \(A\) is the set \(\operatorname{Ker} A = \{x \in \mathbb{C}^m: Ax = 0 \}\)
Suppose \(A \in \mathbb{C}^{m \times n}\) has linearly independent columns. Prove that \(A(A^*A)^{-1}A^*\) is an orthogonal projection onto the column space of \(A\).
For any matrix \(A \in \mathbb{C}^{m \times n}\), prove that \(\operatorname{Ker} (A) = \operatorname{Ker} (A^*A)\). To prove this, you need to prove two inclusions: \(\operatorname{Ker} (A) \subseteq \operatorname{Ker} (A^*A)\) and \(\operatorname{Ker} (A) \supseteq \operatorname{Ker} (A^*A)\). One of these is easy. To prove the other, use the fact that \[\|Ax\|^2 = \langle Ax, Ax \rangle = \langle A^*A x, x \rangle.\]
Find the least squares model for the plane \(z = a+bx+cy\) that best fits the points \((1,1,3), (0,3,6), (2,1,5), (0,0,0)\). Hint: Set this up as a system of 4 equations with 3 unknowns (\(a\), \(b\), and \(c\)). Rewrite the system as a matrix equation and use least squares to find the values of \(a\), \(b\), and \(c\) that work best.
Find the best fit parabola \(y = ax^2 + bx +c\) for the points \((0,1)\), \((1,3)\), \((-1,3)\), and \((3,20)\)