Linear Independence and Basis

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

    2.   (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) \(\left[\begin{array}{r} 1\\ 2 \\ -1\end{array}\right],\left[\begin{array}{r} 2 \\-1 \\ 0\end{array}\right]\)


  2. Find a basis for the set of vectors in \(\mathbb{R}^{2}\) on the line \(y = 3x\).   solution


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

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


  4. Find a basis for the space spanned by the given five vectors.   solution

    5. \(\left[\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right],\left[\begin{array}{r}-2 \\ 1 \\ -1 \\ 1\end{array}\right],\left[\begin{array}{r}6 \\ -1 \\ 2 \\ -1\end{array}\right],\left[\begin{array}{r}5 \\ -3 \\ 3 \\ -4\end{array}\right],\left[\begin{array}{r}0 \\ 3 \\ -1 \\ 1\end{array}\right]\)


  5. Let \(\overrightarrow{v}_{1}=\left[\begin{array}{r}7 \\ 4 \\ -9 \\ -5\end{array}\right],  \: \overrightarrow{v}_{2}=\left[\begin{array}{r}4 \\ -7 \\ 2 \\ 5\end{array}\right], \: \overrightarrow{v}_{3}=\left[\begin{array}{r}1 \\ -5 \\ 3 \\ 4\end{array}\right] .\) It can be verified that \(\overrightarrow{v}_{1}-3 \overrightarrow{v}_{2}+5 \overrightarrow{v}_{3}=\overrightarrow{0} .\) Use this information to find a basis for \(H=\operatorname{Span}\left\{\overrightarrow{v}_{1}, \overrightarrow{v}_{2}, \overrightarrow{v}_{3}\right\}\).   solution


  6. The first four Hermite polynomials are \(1, 2t, -2+4t^2,-12t+8t^3\). These polynomials arise naturally in the study of certain important differential equations in mathematical physics. Show that the first four Hermite polynomials form a basis of \(\mathbb{P}_{3}\).   solution



  7. The first four Laguerre polynomials are \(1, 1-t, 2-4t+t^2,6-18t+9t^2-t^3\). Show that these polynomials form a basis of \(\mathbb{P}_{3}\).  solution


  8. Assume that the matrix A is row equivalent to B. Without calculations, list rank A and dim Nul A. Then find bases for Col A, Row A, and Nul A.   solution

    \( A=\left[\begin{array}{rrrrr}1 & -3 & 4 & -1 & 9 \\ -2 & 6 & -6 & -1 & -10 \\ -3 & 9 & -6 & -6 & -3 \\ 3 & -9 & 4 & 9 & 0\end{array}\right],       B=\left[\begin{array}{rrrrr}1 & -3 & 0 & 5 & -7 \\ 0 & 0 & 2 & -3 & 8 \\ 0 & 0 & 0 & 0 & 5 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\)


  9. Problems related to Rank, Col A, Nul A, Row A.    solution