Define the coordinate vector of a vector with respect to a basis, and its role in linear transformat...

To define the coordinate vector of a vector with respect to a basis, let's first establish what a basis is. A basis of a vector space is a set of vectors that are linearly independent and span the entire space. This means that any vector in the space can be expressed as a unique linear combination of the basis vectors.

Let's denote a basis for an \( n \)-dimensional vector space \( V \) as \( B = \{ \mathbf{b}_1, \mathbf{b}_2, \ldots, \mathbf{b}_n \} \), where each \( \mathbf{b}_i \) is a vector in \( V \).

Now, consider a vector \( \mathbf{v} \) in \( V \). The coordinate vector of \( \mathbf{v} \) with respect to the basis \( B \), denoted as \( [\mathbf{v}]_B \), is a column vector whose entries are the coefficients required to express \( \mathbf{v} \) as a linear combination of the basis vectors. In other words, if

\[ \mathbf{v} = c_1\mathbf{b}_1 + c_2\mathbf{b}_2 + \ldots + c_n\mathbf{b}_n, \]

then

\[ [\mathbf{v}]_B = \begin{bmatrix} c_1 \\ c_2 \\ \vdots \\ c_n \end{bmatrix}. \]

Here's a step-by-step process to find the coordinate vector \( [\mathbf{v}]_B \):

1. Express the Vector as a Linear Combination: Write the vector \( \mathbf{v} \) as a linear combination of the basis vectors \( \mathbf{b}_1, \mathbf{b}_2, \ldots, \mathbf{b}_n \).

2. Solve for Coefficients: Solve the resulting system of linear equations to find the coefficients \( c_1, c_2, \ldots, c_n \).

3. Form the Coordinate Vector: Arrange the coefficients in a column matrix to form the coordinate vector \( [\mathbf{v}]_B \).

The role of the coordinate vector in linear transformations is crucial. A linear transformation \( T: V \rightarrow W \) between two vector spaces can be represented by a matrix \( A \) with respect to the bases of \( V \) and \( W \). If \( [\mathbf{v}]_B \) is the coordinate vector of \( \mathbf{v} \) in \( V \) with respect to basis \( B \), and \( [\mathbf{w}]_C \) is the coordinate vector of \( \mathbf{w} \) in \( W \) with respect to basis \( C \), then the action of the linear transformation on \( \mathbf{v} \) can be described by the matrix equation:

\[ [\mathbf{w}]_C = A[\mathbf{v}]_B. \]

This equation tells us that to find the image of \( \mathbf{v} \) under the transformation \( T \), we can multiply the matrix \( A \) representing \( T \) by the coordinate vector \( [\mathbf{v}]_B \), resulting in the coordinate vector \( [\mathbf{w}]_C \) of the transformed vector \( \mathbf{w} \) in \( W \).

In summary, the coordinate vector of a vector with respect to a basis is a representation of that vector in terms of the basis vectors. It plays a fundamental role in linear algebra, particularly in the context of linear transformations, where it allows us to apply transformations using matrix multiplication.