2025-09-26

  1. If \(A\) is an invertible matrix, then which of the following statements is false?

  1. \(det \left( A \right) = 0\)

  2. \(A x = b\) has a unique solution for any vector \(b\)

  3. \(A^T\) is invertible

  4. The nullspace of \(A\) only contains the vector \(x = 0\)

Answera

2. Suppose I have used the Gram-Schmidt process to decompose a square matrix \(A\) into an orthonormal matrix \(Q\) and a upper triangular matrix \(R\) such that \(A = Q R\). Which of the following statements is true?

  1. \(R^T R = I\)

  2. \(Q\) is a square matrix

  3. \(A^{-1} = Q^T R^{-1}\)

  4. \(q_i \cdot q_j = 1\) for any two columns of \(Q\)

Answerb

3. Sketch a diagram showing the decomposition of a 2D vector \(a\) into its projection onto a unit vector \(v\), given by \(\left( a \cdot v \right) v\), and the projection orthogonal to the unit vector \(v\), given by \(a - \left( a \cdot v \right) v\). Assume that \(a\) is not parallel or orthogonal to \(v\).

AnswerSee Wikipedia for an example.