To understand how different bases connect in coordinate representation, it’s good to know what a basis is in vector spaces.

A basis is a group of vectors that are independent from each other and can cover a whole vector space. When you have a vector space, let's call it $V$, you can express any vector in that space as a combination of the basis vectors. But remember, this way of expressing a vector depends on the basis you choose.

Think of a vector space like $\mathbb{R}^n$. Imagine you have two bases:

• $B_1 = \{ \mathbf{b_1}, \mathbf{b_2}, \dots, \mathbf{b_n} \}$
• $B_2 = \{ \mathbf{c_1}, \mathbf{c_2}, \dots, \mathbf{c_n} \}$

You can express a vector $\mathbf{v}$ in both bases. Let’s call its coordinates in basis $B_1$ as $\mathbf{v}_{B_1}$ and in basis $B_2$ as $\mathbf{v}_{B_2}$.

To connect these two ways of representing the vector, you need to change the basis. The important part here is the change of basis matrix, usually referred to as $P$. This matrix is built using the coordinates of the new basis vectors (the $B_2$ vectors) expressed in terms of the original basis (the $B_1$ vectors).

To put it simply, if you write each vector from $B_2$ using vectors from $B_1$, then the change of basis matrix $P$ looks like this:

$$P = \begin{bmatrix} \text{Coord}(\mathbf{c_1} \text{ in } B_1) & \text{Coord}(\mathbf{c_2} \text{ in } B_1) & \cdots & \text{Coord}(\mathbf{c_n} \text{ in } B_1) \end{bmatrix}$$

With this matrix, switching between the two ways of showing the vector becomes pretty easy. You can use:

$$\mathbf{v}_{B_2} = P \cdot \mathbf{v}_{B_1}$$

If you want to go back from $B_2$ to $B_1$, you need the inverse of the change of basis matrix:

$$\mathbf{v}_{B_1} = P^{-1} \cdot \mathbf{v}_{B_2}$$

This whole process highlights something important about linear transformations. Since changing the basis is a linear transformation itself, combining transformations with different bases keeps things clear and organized.

Now, think about what this means. Different bases give you new ways to look at the same vector or shape. For example, one basis might make certain computations easier, especially when symmetry is involved, while another might show the features of a specific problem better. When you change bases, you’re really looking at the same thing from another angle.

Keep in mind, no matter which basis you pick, the size of the vector space stays the same. Every basis in an $n$-dimensional space will always have $n$ vectors. Changing the basis doesn’t change the space itself; it just gives you a different view of it. This is a lot like giving directions in different systems—like using latitude and longitude instead of a local map. Each system has its own role and is useful depending on what you need.

When you work on problems in linear algebra, understanding how to change bases helps with grasping linear transformations. Many changes—like turning, resizing, and shifting—become easier to understand in different bases. So, picking the right basis can not only make calculations simpler but can also help you find clearer solutions to tricky problems in higher dimensions.

In short, different bases connect through clear processes involving change of basis matrices and linear transformations. These methods can change our view and help us understand vector spaces better. So, make use of these tools! They can help you solve a wider array of problems in linear algebra and beyond.