## Subsection3.1.3What you will learn

This chapter focuses on the QR factorization as a method for computing an orthonormal basis for the column space of a matrix.

Upon completion of this week, you should be able to

• Relate Gram-Schmidt orthogonalization of vectors to the QR factorization of a matrix.

• Show that Classical Gram-Schmidt and Modified Gram-Schmidt yield the same result (in exact arithmetic).

• Compare and contrast the Classical Gram-Schmidt and Modified Gram-Schmidt methods with regard to cost and robustness in the presence of roundoff error.

• Derive and explain the Householder transformations (reflections).

• Decompose a matrix to its QR factorization via the application of Householder transformations.

• Analyze the cost of the Householder QR factorization algorithm.

• Explain why Householder QR factorization yields a matrix $Q$ with high quality orthonormal columns, even in the presence of roundoff error.