The Analysis of Data, volume 1

Linear Algebra: Basic Definitions

C.1. Basic Definitions

  1. What is the column space of the matrix $\begin{pmatrix} 1&1&1\\0&1&1\\ 0 & 0 & 0\end{pmatrix}$? (Identify a set of orthogonal vectors that span it).
  2. What is the column space, null space, and rank of the matrix whose entries are all ones.
  3. Find the eigenvalues and eigenvectors of the matrix $\begin{pmatrix} 2&1\\1&2\end{pmatrix}$.
  4. Consider the vectors $(1,1,0,0)$ and $(0,0,1,1)$. Show that they are linearly independent, and find two additional vectors that together with the two vectors above form a basis for $\R^4$.
  5. Find a matrix of size $4\times 4$ whose null space is spanned by the vector $(1,0,0,0)$.
  6. Compute analytically the spectral decomposition and the SVD of the matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$.