Answer to Question #246783 in Linear Algebra for SOOSAN FIONA A

Question #246783
1. Find the dimension of the subspace spanned by the following vectors in V3(R). (1, 0, 2), (2, 0, 1), (1, 0, 1).
1
Expert's answer
2021-10-06T13:33:27-0400

Solution:We know that Basis of vector space V is a linearly independent set that spans V. 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. Number of nonzero rows are the dimensions of the subspace spanned by the given vectors. We form a matrix whose rows are given by these vectors and reduce  it to echelon form[102201101]Apply R22R1R2    [102003101]Apply R3R1 R3    [102003001]Apply (R3/3) R3    [102001001]Apply R3+R2R3    [102001000]Here nonzero vectors are 2.2 is the dimension of the subspace spanned by the given vectors inV3(R).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).


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