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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1995 • Question 1

Question

Math 6420 • PDE Fall 1995 • Question 1

Let \(A\) be a real \(m\times n\) matrix with adjoint \(A^*\).

  1. Prove that if \(R(A)\) and \(N(A)\) denote the range and nullspace of \(A\) respectively, then \[ R(A)\oplus N(A^*)=\mathbb {R}^m. \] (\(\oplus \) means orthogonal direct sum.)
  2. Using the result of (a), state and prove Fredholm’s Alternative Theorem.

Raw LaTeX

Let \(A\) be a real \(m\times n\) matrix with adjoint \(A^*\).
\begin{enumerate}
\item[(a)] Prove that if \(R(A)\) and \(N(A)\) denote the range and nullspace of \(A\)
respectively, then
\[
  R(A)\oplus N(A^*)=\mathbb{R}^m.
\]
(\(\oplus\) means orthogonal direct sum.)
\item[(b)] Using the result of (a), state and prove Fredholm's Alternative Theorem.
\end{enumerate}

Topic Tags

Adjoint Operators and Riesz Representation Fredholm Alternative Hilbert Spaces and Orthogonality Linear Programming and Optimization