Change of basis matrix  Alternate coordinate systems (bases)  Linear Algebra  Khan Academy  Summary and Q&A
TL;DR
Understanding how to change between bases and coordinate representations using a change of basis matrix.
Questions & Answers
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.