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 = 1 23
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 = 1 23
Solution
The given basis B is {(1, 0, 1), (1, 1, 0), (0, 1, 1)} and the coordinate matrix of x relative to this basis is [x]B = [1, 2, 3].
To find the coordinate matrix of x relative to the standard basis, we need to express x as a linear combination of the standard basis vectors.
The standard basis for R^3 is {(1, 0, 0), (0, 1, 0), (0, 0, 1)}.
We can express x as a linear combination of the vectors in B as follows:
x = 1*(1, 0, 1) + 2*(1, 1, 0) + 3*(0, 1, 1)
= (1 + 2 + 0, 0 + 2 + 3, 1 + 0 + 3)
= (3, 5, 4)
So, the coordinate matrix of x relative to the standard basis is [3, 5, 4].
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