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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2010 • Question 1

Question

Math 6420 • PDE Fall 2010 • Question 1

  1. State and prove the Fredholm Alternative Theorem for a linear transformation \(A:\mathbb {R}^n\to \mathbb {R}^m\).
  2. Find and give a graphical interpretation of the pseudo-inverse of the matrix \[ A= \begin {pmatrix} 1 & 1\\ 2 & 2 \end {pmatrix}. \]

Raw LaTeX

\begin{enumerate}
\item[(a)] State and prove the Fredholm Alternative Theorem for a linear transformation
\(A:\mathbb{R}^n\to\mathbb{R}^m\).

\item[(b)] Find and give a graphical interpretation of the pseudo-inverse of the matrix
\[
  A=
  \begin{pmatrix}
    1 & 1\\
    2 & 2
  \end{pmatrix}.
\]
\end{enumerate}

Topic Tags

Conformal Mapping and Bilinear Transformations Fredholm Alternative Linear Programming and Optimization Moore-Penrose Pseudoinverse