Test questions on unique representation of vectors in a basis

📚 Quick Study Guide

• 📐 Basis Definition: A basis for a vector space is a set of linearly independent vectors that span the entire space. This means any vector in the space can be written as a unique linear combination of the basis vectors.
• ➕ Linear Combination: A linear combination of vectors $v_1, v_2, ..., v_n$ is an expression of the form $c_1v_1 + c_2v_2 + ... + c_nv_n$, where $c_1, c_2, ..., c_n$ are scalars.
• 🧮 Unique Representation Theorem: For a vector space $V$ with basis $B = \{v_1, v_2, ..., v_n\}$, every vector $v$ in $V$ can be written as a unique linear combination of the basis vectors in $B$.
• 🎯 Coordinates: If $v = c_1v_1 + c_2v_2 + ... + c_nv_n$, then the scalars $c_1, c_2, ..., c_n$ are called the coordinates of $v$ with respect to the basis $B$, written as $[v]_B = \begin{bmatrix} c_1 \\ c_2 \\ ... \\ c_n \end{bmatrix}$.
• 🧭 Change of Basis: Changing from one basis to another involves finding the transformation matrix that converts coordinates from one basis to the other.

🧪 Practice Quiz

1. Which of the following statements is true about a basis for a vector space?
   1. A) It must contain all vectors in the vector space.
   2. B) It is a set of linearly dependent vectors that span the space.
   3. C) It is a set of linearly independent vectors that span the space.
   4. D) It is a set of vectors that may or may not span the space.
2. Given a vector $v = 3v_1 - 2v_2$, where $\{v_1, v_2\}$ is a basis for $V$, what are the coordinates of $v$ with respect to this basis?
   1. A) $\begin{bmatrix} -2 \\ 3 \end{bmatrix}$
   2. B) $\begin{bmatrix} 3 \\ -2 \end{bmatrix}$
   3. C) $\begin{bmatrix} 2 \\ -3 \end{bmatrix}$
   4. D) $\begin{bmatrix} -3 \\ 2 \end{bmatrix}$
3. If a set of vectors is linearly dependent, can it be a basis for a vector space?
   1. A) Yes, always.
   2. B) Only if it spans the space.
   3. C) No, never.
   4. D) Only if it contains the zero vector.
4. Suppose $\{v_1, v_2\}$ is a basis for $R^2$. If $v = av_1 + bv_2$, what can be said about the values of $a$ and $b$?
   1. A) They are any real numbers.
   2. B) They are unique for the given vector $v$.
   3. C) They must be positive integers.
   4. D) They must be zero.
5. Which of the following sets of vectors forms a basis for $R^2$?
   1. A) $\{(1, 0), (0, 1), (1, 1)\}$
   2. B) $\{(1, 1), (2, 2)\}$
   3. C) $\{(1, 0), (0, 1)\}$
   4. D) $\{(0, 0), (1, 1)\}$
6. Let $B = \{v_1, v_2\}$ be a basis for a vector space $V$. If $w = 2v_1 - v_2$, what is the coordinate vector $[w]_B$?
   1. A) $\begin{bmatrix} -1 \\ 2 \end{bmatrix}$
   2. B) $\begin{bmatrix} 1 \\ -2 \end{bmatrix}$
   3. C) $\begin{bmatrix} 2 \\ -1 \end{bmatrix}$
   4. D) $\begin{bmatrix} -2 \\ 1 \end{bmatrix}$
7. If a vector space $V$ has a basis with $n$ vectors, what is the dimension of $V$?
   1. A) $2n$
   2. B) $n^2$
   3. C) $n$
   4. D) $n-1$