Change of basis matrix | Alternate coordinate systems (bases) | Linear Algebra | Khan Academy | Summary and Q&A

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November 11, 2009
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Change of basis matrix | Alternate coordinate systems (bases) | Linear Algebra | Khan Academy

TL;DR

Understanding how to change between bases and coordinate representations using a change of basis matrix.

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Q: What is the purpose of a change of basis matrix?

A change of basis matrix allows us to convert the coordinate representation of a vector between different bases. It helps us change bases and operate in different coordinate systems.

Q: How can we represent a vector in its standard coordinates?

To represent a vector in its standard coordinates, we can multiply the vector's coordinates with respect to a given basis by the change of basis matrix with the basis vectors as columns.

Q: How can we determine the coordinates of a vector with respect to a given basis?

By solving the equation C times the vector of coordinates with respect to the basis equals the vector itself, where C is the change of basis matrix.

Q: Can we change between bases if we know the standard coordinate representation of a vector?

Yes, we can determine the coordinates of a vector with respect to a given basis by solving the equation C times the vector of standard coordinates equals the vector, where C is the change of basis matrix.

Summary & Key Takeaways

• When we have a basis B and a vector a, the coordinates of a with respect to B represent the weights of the basis vectors needed to create a linear combination that equals a.

• This concept can be represented with a matrix C, where the column vectors are the basis vectors of B. The matrix C multiplied by the vector of coordinates of a with respect to B will equal a.

• The change of basis matrix helps us change bases and can be used to convert the coordinate representation of a vector from one basis to the standard basis, or vice versa.