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 non−zero 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∴⎣⎡121000211⎦⎤Apply R2−2R1→R2 ⎣⎡1010002−31⎦⎤Apply R3−R1→ R3 ⎣⎡1000002−3−1⎦⎤Apply (R3/−3)→ R3 ⎣⎡10000021−1⎦⎤Apply R3+R2→R3 ⎣⎡100000210⎦⎤Here non−zero vectors are 2.∴2 is the dimension of the subspace spanned by the given vectors inV3(R).
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