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Exercise 101

Avg difficulty 2.5

For the following sets of vectors, find a basis for the subspace spanned by the vectors, and find the dimension of the subspace.

  1. [12],[11]\begin{bmatrix} 1 \\ 2 \end{bmatrix} , \quad \begin{bmatrix} 1 \\ 1 \end{bmatrix}
  2. [111],[222],[112]\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} , \quad \begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix} , \quad \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}
  3. [531],[5−15],[−13−4]\begin{bmatrix} 5 \\ 3 \\ 1 \end{bmatrix} , \quad \begin{bmatrix} 5 \\ -1 \\ 5 \end{bmatrix} , \quad \begin{bmatrix} -1 \\ 3 \\ -4 \end{bmatrix}
  4. [224],[223],[44−3]\begin{bmatrix} 2 \\ 2 \\ 4 \end{bmatrix} , \quad \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} , \quad \begin{bmatrix} 4 \\ 4 \\ -3 \end{bmatrix}
  5. [10],[20],[30]\begin{bmatrix} 1 \\ 0 \end{bmatrix} , \quad \begin{bmatrix} 2 \\ 0 \end{bmatrix} , \quad \begin{bmatrix} 3 \\ 0 \end{bmatrix}
  6. [100],[200],[012]\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , \quad \begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix} , \quad \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}

Lebl, Notes on Diffy Qs, §10.4 Subspaces, dimension, and the kernel, exercise 101

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Adapted from Notes on Diffy Qs © 2008–2026 Jiří Lebl. Licensed under CC BY-SA 4.0. Source: Notes on Diffy Qs: Differential Equations for Engineers. Imported at revision 658bcae.