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Linear Algebra

Worksheet 9

  1. Compute the LU factorization with partial pivoting of \(A = \begin{pmatrix} F & T & F & F \\ T & T & T & F \\ F & T & T & T \end{pmatrix}\)

  2. Which the following are vector spaces and which are not:
    • The subset of points in $\mathbb{R}^2$ lying on the graph $y = x^2$.
    • The set of orthogonal matrices.
    • The set of lower triangular matrices.
    • The set of symmetric matrices. A matrix is called symmetric if $A = A^T$.
  3. Assume that $V_1$ and $V_2$ are $k$-vector spaces. Consider the set \(V_1 \times V_2 := \lbrace (v_1,v_2) \mid v_1 \in V_1, v_2 \in V_2 \rbrace\) with $+$ given by \((v_1,v_2) + (v_1^\prime,v_2^\prime) := (v_1+v_1^\prime, v_2+v_2^\prime)\) and $\cdot$ given by \(c \cdot (v_1, v_2) := (c\cdot v_1,c\cdot v_2).\) Prove that $V_1 \times V_2$ with these $+$ and $\cdot$ is a $k$-vector space. It is called the product of $V_1$ and $V_2$.