Question

1. Find the dimension of the subspace spanned by the following vectors in V3(R). (1, 0, 2), (2, 0, 1), (1, 0, 1).

Answer 1

$Solution: \\We ~ know~ that~ Basis ~of ~vector ~space ~V ~is~ a ~linearly~ independent ~set~ that~ spans~ V. \\ \therefore ~dimension~ of ~V = Card(basis ~of~ V) \\Now ~to ~check ~ linearly ~independent~, we ~need ~to~ write~ given ~vectors~ in ~matrix ~ \\form ~and ~reduce~ it ~to ~row ~echelon ~form. \\ \therefore ~Number ~of ~non- zero~ rows ~are~ the~ dimensions ~of~ the ~subspace~ spanned ~\\by ~the~ given~ vectors. \\ \therefore~ We~ form~ a~ matrix ~whose ~rows~ are~ given ~by ~these ~vectors ~and~ reduce~ ~\\it ~to~ echelon~ form \\ \therefore \begin{bmatrix} 1 & 0 & 2\\ 2 & 0 & 1\\ 1 & 0 & 1 \end{bmatrix} \\Apply~ R_2 -2R_1 \rightarrow R_2 \\ ~~~~\begin{bmatrix} 1 & 0 & 2\\ 0 & 0 & -3\\ 1 & 0 & 1 \end{bmatrix} \\ Apply ~R_3 -R_1 \rightarrow ~R_3 \\ ~~~~\begin{bmatrix} 1 & 0 & 2\\ 0 & 0 & -3\\ 0 & 0 & -1 \end{bmatrix} \\Apply~(R_3/-3) \rightarrow~R_3 \\~~~~\begin{bmatrix} 1 & 0 & 2\\ 0 & 0 & 1\\ 0 & 0 & -1 \end{bmatrix} \\Apply~R_3+R_2 \rightarrow R_3 \\~~~~\begin{bmatrix} 1 & 0 & 2\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{bmatrix} \\ Here~non-zero~ vectors ~are~2. \\ \therefore 2 ~is ~the~ dimension~ of ~the~ subspace~ spanned~ by ~the~given ~vectors~ in V3(R).$