Linear Algebra Example Problems - Vector Space Basis Example #1
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In this video we determine if a set of given vectors is a basis for R3.
We know that in general, a basis for Rn requires n linearly independent vectors. Since we're given 3 vectors in this problem, we require these 3 vectors to be linearly independent if they are to form a basis for R3.
Two different methods are used to check for linear independence of the vectors. In the first, we construct a matrix and perform row operations to show that we obtain a pivot in each column. This implies that the only solution to Ax = 0 is the trivial solution (i.e. x = 0) and thus the vectors are independent.
In the second method we compute the determinant of the matrix. Since the determinant is non-zero, the vectors are independent.
Since we've shown that the three vectors are linearly independent, then they form a basis for R3.
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