Solved examples: finding the dimension of a vector space

📚 Quick Study Guide

🔍 Definition of Vector Space: A vector space is a set that is closed under vector addition and scalar multiplication. 💡 Definition of Dimension: The dimension of a vector space $V$ is the number of vectors in a basis for $V$. All bases for $V$ have the same number of vectors. 📝 Finding a Basis: To find a basis, find a set of linearly independent vectors that span the vector space. ➗ Linear Independence: Vectors $v_1, v_2, ..., v_n$ are linearly independent if the equation $c_1v_1 + c_2v_2 + ... + c_nv_n = 0$ has only the trivial solution $c_1 = c_2 = ... = c_n = 0$. 📐 Spanning Set: A set of vectors {$v_1, v_2, ..., v_n$} spans a vector space $V$ if every vector in $V$ can be written as a linear combination of $v_1, v_2, ..., v_n$. ➕ Dimension of $\mathbb{R}^n$: The dimension of the vector space $\mathbb{R}^n$ is $n$. 🗺️ Examples: The dimension of the vector space of all $m \times n$ matrices is $mn$.

Practice Quiz

1. What is the dimension of the vector space $\mathbb{R}^3$? A. 1 B. 2 C. 3 D. 4 2. What is the dimension of the vector space of all $2 \times 2$ matrices? A. 2 B. 3 C. 4 D. 5 3. Let $V = \{ (x, y) \in \mathbb{R}^2 : x + y = 0 \}$. What is the dimension of $V$? A. 0 B. 1 C. 2 D. Infinite 4. What is the dimension of the vector space spanned by the vectors $(1, 0, 0)$, $(0, 1, 0)$, and $(0, 0, 1)$? A. 1 B. 2 C. 3 D. 4 5. Let $W = \{ (x, y, z) \in \mathbb{R}^3 : x + y + z = 0 \}$. What is the dimension of $W$? A. 1 B. 2 C. 3 D. 0 6. What is the dimension of the vector space consisting of all polynomials of degree at most 2? A. 1 B. 2 C. 3 D. 4 7. What is the dimension of the zero vector space, $\{\vec{0}\}$? A. 0 B. 1 C. Infinite D. Undefined

Click to see Answers

1. C
2. C
3. B
4. C
5. B
6. C
7. A