Probability
The Analysis of Data, volume 1
Linear Algebra: Basic Definitions
$
\def\P{\mathsf{\sf P}}
\def\E{\mathsf{\sf E}}
\def\Var{\mathsf{\sf Var}}
\def\Cov{\mathsf{\sf Cov}}
\def\std{\mathsf{\sf std}}
\def\Cor{\mathsf{\sf Cor}}
\def\R{\mathbb{R}}
\def\c{\,|\,}
\def\bb{\boldsymbol}
\def\diag{\mathsf{\sf diag}}
$
C.1. Basic Definitions
- 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).
- What is the column space, null space, and rank of the matrix whose entries are all ones.
- Find the eigenvalues and eigenvectors of the matrix $\begin{pmatrix} 2&1\\1&2\end{pmatrix}$.
- 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 $\R^4$.
- Find a matrix of size $4\times 4$ whose null space is spanned by the vector $(1,0,0,0)$.
- Compute analytically the spectral decomposition and the SVD of the matrix
$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$.