Worked Examples: Converting Vectors to Coordinate Vectors in R^n

📚 Quick Study Guide

🔍 A vector in $R^n$ can be represented as a linear combination of basis vectors. 💡 The coordinate vector of a vector $v$ with respect to a basis $B = {b_1, b_2, ..., b_n}$ is the vector of coefficients needed to express $v$ as a linear combination of the basis vectors. 📝 To find the coordinate vector $[v]_B$, solve the equation $v = c_1b_1 + c_2b_2 + ... + c_nb_n$ for the scalars $c_1, c_2, ..., c_n$. These scalars form the coordinate vector $[v]_B = \begin{bmatrix} c_1 \\ c_2 \\ ... \\ c_n \end{bmatrix}$. ➕ Changing the basis will change the coordinate vector, even though the vector itself remains the same. 📏 The coordinate vector depends on the order of the basis vectors in $B$. 🔑 The transition matrix is key for converting coordinate vectors from one basis to another.

Practice Quiz

1. What is a coordinate vector?
   1. A vector with coordinates (x, y, z).
   2. The vector of coefficients when expressing a vector as a linear combination of basis vectors.
   3. A vector that is orthogonal to the original vector.
   4. A vector with a magnitude of 1.
2. Given the standard basis $B = {\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}}$ in $R^2$, what is the coordinate vector of $v = \begin{bmatrix} 3 \\ -2 \end{bmatrix}$ with respect to $B$?
   1. $\begin{bmatrix} -2 \\ 3 \end{bmatrix}$
   2. $\begin{bmatrix} 3 \\ -2 \end{bmatrix}$
   3. $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$
   4. $\begin{bmatrix} 0 \\ 0 \end{bmatrix}$
3. Let $B = {\begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ -1 \end{bmatrix}}$ be a basis for $R^2$. Find the coordinate vector of $v = \begin{bmatrix} 2 \\ 4 \end{bmatrix}$ with respect to $B$.
   1. $\begin{bmatrix} 3 \\ -1 \end{bmatrix}$
   2. $\begin{bmatrix} 3 \\ 1 \end{bmatrix}$
   3. $\begin{bmatrix} 1 \\ 3 \end{bmatrix}$
   4. $\begin{bmatrix} -1 \\ 3 \end{bmatrix}$
4. What is the importance of the order of basis vectors when determining coordinate vectors?
   1. The order doesn't matter.
   2. Different orders result in the same coordinate vector.
   3. Different orders may result in different coordinate vectors.
   4. Only the first basis vector matters.
5. Which of the following is NOT a step to find the coordinate vector of a vector $v$ with respect to a basis $B = {b_1, b_2, ..., b_n}$?
   1. Express $v$ as a linear combination of the basis vectors.
   2. Solve for the coefficients in the linear combination.
   3. Form a vector with the coefficients.
   4. Calculate the determinant of the basis vectors.
6. If $[v]_B = \begin{bmatrix} 2 \\ -1 \end{bmatrix}$ with respect to the basis $B = {\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}}$, what is the vector $v$?
   1. $\begin{bmatrix} 0 \\ 0 \end{bmatrix}$
   2. $\begin{bmatrix} 2 \\ -1 \end{bmatrix}$
   3. $\begin{bmatrix} -1 \\ 2 \end{bmatrix}$
   4. $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$
7. Given the basis $B = {\begin{bmatrix} 2 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix}}$ and the vector $v = \begin{bmatrix} 5 \\ 3 \end{bmatrix}$, find the coordinate vector $[v]_B$.
   1. $\begin{bmatrix} 2 \\ 1 \end{bmatrix}$
   2. $\begin{bmatrix} 1 \\ 2 \end{bmatrix}$
   3. $\begin{bmatrix} 3 \\ -1 \end{bmatrix}$
   4. $\begin{bmatrix} -1 \\ 3 \end{bmatrix}$