Subspace of \(\mathbb{R}^n\)

  1. Determine which sets are bases for \(\mathbb{R}^2\) or \(\mathbb{R}^3\). Justify each answer.  solution

    2. (a) \(\left[\begin{array}{r} 3 \\ 1 \\ -2 \end{array}\right],\left[\begin{array}{r} 1 \\ -7 \\ 5 \end{array}\right] \)   (b) \(\left[\begin{array}{r}8 \\ -6 \end{array}\right],\left[\begin{array}{r} -4 \\ 3\end{array}\right]\)   (c) \(\left[\begin{array}{r} 1 \\ 5 \\ -2 \end{array}\right],\left[\begin{array}{r}5 \\ -7 \\ 4 \end{array}\right],\left[\begin{array}{r}2 \\ -3 \\ 5 \end{array}\right], \left[\begin{array}{r}1 \\ 3 \\ -2 \end{array}\right]\)  (d) \(\left[\begin{array}{r}0 \\ 1 \\ -2 \end{array}\right],\left[\begin{array}{r}5 \\ -7 \\ 4 \end{array}\right],\left[\begin{array}{r}6 \\ 3 \\ 5 \end{array}\right]\)


  2. Determine if the vectors form a basis for \(\mathbb{R}^3\).   solution

    3.   (a) \(\left[\begin{array}{l} 1\\ 1 \\ 0\end{array}\right],\left[\begin{array}{l} 2 \\ 3 \\ 0\end{array}\right],\left[\begin{array}{l}{1} \\ 2 \\ {3}\end{array}\right]\)      (b) \(\left[\begin{array}{r} 1\\ 2 \\ -1\end{array}\right],\left[\begin{array}{l} 2 \\1 \\ 0\end{array}\right],\left[\begin{array}{r} 5 \\ 4\\ {-1}\end{array}\right]\)     (c) \(\left[\begin{array}{r} 1\\ 2 \\ -1\end{array}\right],\left[\begin{array}{r} 2 \\-1 \\ 0\end{array}\right]\)


  3. Find a basis for the null space of the matrix.   solution

    4. \(A=\left[\begin{array}{rrrrr}1 & 5 & -4 & -3 & 1 \\ 0 & 1 & -2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\)  


  4. Find a basis for the column space of the matrix.  solution

    5. \(A=\left[\begin{array}{rrrrr} 1 & 0 & -4 & -3 & 0 \\ 0 & 1 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 &1 \\ 0 & 0 & 0 & 0 &0 \end{array}\right]\)


  5. Assume that \(A\) is row equivalent to \(B\). Find bases for Nul A and Col A.  solution

    6. \(A=\left[\begin{array}{rrrrr}
    1 & 2 & -5 & 11 & -3 \\
    2 & 4 & -5 & 15 & 2 \\
    1 & 2 & 0 & 4 & 5 \\
    3 & 6 & -5 & 19 & -2
    \end{array}\right],  \quad   B=\left[\begin{array}{ccccr}1 & 2 & 0 & 4 & 5 \\ 0 & 0 & 5 & -7 & 8 \\ 0 & 0 & 0 & 0 & -9 \\ 0 & 0 & 0 & 0 & 0\end{array}\right] \)