Examples of dimensions for common vector spaces (R^n, P_n, M_mn)

📚 Quick Study Guide

🔍 Vector Space Dimension: The dimension of a vector space $V$ is the number of vectors in a basis for $V$. A basis is a set of linearly independent vectors that span the entire vector space. 💡 Dimension of $\mathbb{R}^n$: The dimension of the vector space $\mathbb{R}^n$ (the set of all $n$-tuples of real numbers) is $n$. A standard basis for $\mathbb{R}^n$ is the set of vectors {$e_1, e_2, ..., e_n$}, where $e_i$ has a 1 in the $i$-th position and 0s elsewhere. 📝 Dimension of $P_n$: The dimension of the vector space $P_n$ (the set of all polynomials of degree at most $n$) is $n+1$. A standard basis for $P_n$ is {1, $x$, $x^2$, ..., $x^n$}. ➗ Dimension of $M_{m,n}$: The dimension of the vector space $M_{m,n}$ (the set of all $m \times n$ matrices with real entries) is $m \times n$. A standard basis for $M_{m,n}$ consists of matrices with a single 1 in one entry and 0s elsewhere.

Practice Quiz

1. What is the dimension of the vector space $\mathbb{R}^5$?

   1. A) 4
   2. B) 5
   3. C) 6
   4. D) 25

2. What is the dimension of the vector space $P_3$ (polynomials of degree at most 3)?

   1. A) 3
   2. B) 4
   3. C) 5
   4. D) 6

3. What is the dimension of the vector space $M_{2,3}$ (2x3 matrices)?

   1. A) 2
   2. B) 3
   3. C) 5
   4. D) 6

4. What is the dimension of the vector space $\mathbb{R}^{10}$?

   1. A) 5
   2. B) 10
   3. C) 100
   4. D) 11

5. What is the dimension of the vector space $P_7$ (polynomials of degree at most 7)?

   1. A) 6
   2. B) 7
   3. C) 8
   4. D) 9

6. What is the dimension of the vector space $M_{4,2}$ (4x2 matrices)?

   1. A) 4
   2. B) 2
   3. C) 6
   4. D) 8

7. If a vector space $V$ has a basis with 6 vectors, what is the dimension of $V$?

   1. A) 5
   2. B) 6
   3. C) 7
   4. D) It cannot be determined.