## Subsection2.1.3What you will learn

This week introduces two concepts that have theoretical and practical importance: unitary matrices and the Singular Value Decomposition (SVD).

Upon completion of this week, you should be able to

• Determine whether vectors are orthogonal.

• Compute the component of a vector in the direction of another vector.

• Relate sets of orthogonal vectors to orthogonal and unitary matrices.

• Connect unitary matrices to the changing of orthonormal basis.

• Identify transformations that can be represented by unitary matrices.

• Prove that multiplying with unitary matrices does not amplify relative error.

• Use norms to quantify the conditioning of solving linear systems.

• Prove and interpret the Singular Value Decomposition.

• Link the Reduced Singular Value Decomposition to the rank of the matrix and determine the best rank-k approximation to a matrix.

• Determine whether a matrix is close to being nonsingular by relating the Singular Value Decomposition to the condition number.