Finding New Coordinates in a Different Basis

Given the vector $v = (2, 3)$ in the standard basis, find the new coordinates $v'$ in the basis defined by $u_1 = (1, 2)$ and $u_2 = (3, 3)$.

When dealing with vectors in different bases, one fundamental concept you'll encounter is the change of basis. A basis essentially provides a framework for understanding the vector space, and every vector can be expressed in terms of this framework. In this problem, you're taking a vector from one basis, the standard basis, and translating it into another basis. This requires understanding how to express the given vector in terms of the new basis vectors, often through the use of a transition matrix or by solving a system of linear equations.

The strategy typically involves expressing the original vector as a linear combination of the new basis vectors. This means you are looking for coefficients that when multiplied by the new basis vectors and summed, will recreate the original vector. This can frequently be solved via setting up an equation involving matrices, where the columns represent the new basis vectors and the vector of coefficients represents the unknowns. By solving this as a linear system, you determine the weights or coordinates of the original vector in the new basis layout. Understanding this process is crucial in linear algebra as it allows flexibility in solving problems and interpreting solutions in different ways, lending insight into the intrinsic properties of vector spaces.