Lecture 7.11- Coordinates with Respect to a Basis

# Lecture 7.11- Coordinates with Respect to a Basis - 7.11...

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7.11 Coordinates with Respect to a Basis MATH232 D100 2016-3 Lecture 26 Paul Tupper SFU Burnaby

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The Standard Basis of R n The standard basis of R n is the basis { e 1 , . . . , e n } consisting of the standard unit vectors. Recall that e j is the vector whose j th component is 1 and whose other components are zeroes. Example: The standard basis of R 4 is { e 1 , e 2 , e 3 , e 4 } with
Then any vector can be expressed as a linear combination of the standard basis vectors, for example: 4 2 - 1 3 =

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Now let B = { v 1 , v 2 , v 3 } , with v 1 = (1 , 3 , 1) , v 2 = (1 , - 2 , 2) , and v 3 = (1 , 1 , 1) . B is a basis of R 3 because there are three vectors and they are linearly independent, which you can check by writing the vectors as columns of a matrix M = 1 1 1 3 - 2 1 1 2 1 and noting that
Suppose we wanted to express a vector, say x = (3 , 4 , 2) as a linear combination of the vectors of B ,i.e. This is equivalent to the matrix equation

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We write to indicate that in order to represent our vector x = (3 , 4 , 2) as a linear combination of our basis B = { v

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