Change of Basis in R2

Consider the bases for $\mathbb{R}^2$: $B = \{ [1, 1], [0, 2] \}$ and $C = \{ [0, 1], [1, 2] \}$. If the coordinate vector for $x$ with respect to $B$ is $[3, 1]$, find the coordinate vector for $x$ with respect to $C$.

This problem explores the concept of changing coordinate systems, focusing on vector spaces and the representation of vectors in different bases. In linear algebra, a basis of a vector space is a set of vectors that are linearly independent and span the space. Changing the basis of a vector essentially involves re-expressing the vector with respect to a different set of basis vectors. This is a fundamental concept as it allows for flexibility in solving problems by choosing a basis that simplifies calculations. To approach this problem, one must understand how to convert a vector from one basis to another using a transition matrix.

The transition matrix can be constructed using the complex relationship between the different basis vectors. This involves expressing each vector of one basis as a linear combination of the vectors of the other basis. By understanding these relationships thoroughly, one can construct the necessary transformation to find the new coordinates in the target basis. The ability to change bases efficiently is a critical skill in linear algebra, as it is widely applicable in various mathematical and engineering contexts.

It is particularly useful in simplifying problems or accommodating specific conditions within a given domain. When faced with similar tasks, always begin by analyzing the relationship between the bases and set up the equations that relate the vectors in each basis. This strategic method not only aids in resolving the problem at hand but also enhances overall problem-solving skills in vector space transformations.