Given the coordinate matrix of x relative to a (nonstandard) basis B for Rn, find the coordinate matrix of x relative to the standard basis.B = {(1, 0, 1), (1, 1, 0), (0, 1, 1)},[x]B = 3 12[x]S =
Question
Given the coordinate matrix of x relative to a (nonstandard) basis B for Rn, find the coordinate matrix of x relative to the standard basis.B = {(1, 0, 1), (1, 1, 0), (0, 1, 1)},[x]B = 3 12[x]S =
Solution
The coordinate matrix of x relative to the standard basis can be found by taking a linear combination of the basis vectors in B, using the coordinates of x in B as the coefficients.
The basis vectors in B are (1, 0, 1), (1, 1, 0), and (0, 1, 1). The coordinates of x in B are 3 and 12.
So, we take 3 times the first basis vector, plus 12 times the second basis vector:
3*(1, 0, 1) + 12*(1, 1, 0) = (3, 0, 3) + (12, 12, 0) = (15, 12, 3)
So, the coordinate matrix of x relative to the standard basis S is [15, 12, 3].
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