How to determine the dimension of U+W and U intersect W using examples

📚 Quick Study Guide

• 📏 Dimension: The number of vectors in a basis for a vector space.
• ➕ Sum of Subspaces (U + W): The set of all vectors that can be written as a sum of a vector in U and a vector in W.
• 🤝 Intersection of Subspaces (U ∩ W): The set of all vectors that are in both U and W.
• 🔑 Key Formula: $\dim(U + W) = \dim(U) + \dim(W) - \dim(U \cap W)$
• 💡 Rearrangement: $\dim(U \cap W) = \dim(U) + \dim(W) - \dim(U + W)$
• 📝 Important Note: The dimension of U+W cannot be greater than the dimension of the entire vector space.

Practice Quiz

1. Question 1: Let U and W be subspaces of $\mathbb{R}^5$. If $\dim(U) = 3$ and $\dim(W) = 2$, what is the maximum possible value of $\dim(U \cap W)$?
   1. 0
   2. 1
   3. 2
   4. 3
2. Question 2: Let U and W be subspaces of a vector space V. If $\dim(U) = 4$, $\dim(W) = 3$, and $\dim(U + W) = 6$, what is $\dim(U \cap W)$?
   1. 0
   2. 1
   3. 2
   4. 3
3. Question 3: If U and W are subspaces of $\mathbb{R}^4$ with $\dim(U) = 2$ and $\dim(W) = 2$, and $U \cap W = {\vec{0}}$, what is $\dim(U + W)$?
   1. 1
   2. 2
   3. 3
   4. 4
4. Question 4: Let U and W be subspaces of V such that $\dim(U) = 5$, $\dim(W) = 3$, and $\dim(V) = 7$. What is the minimum possible value for $\dim(U \cap W)$?
   1. 0
   2. 1
   3. 2
   4. 3
5. Question 5: Suppose U and W are subspaces of $\mathbb{R}^6$, with $\dim(U) = 4$ and $\dim(W) = 4$. Which of the following is NOT a possible value for $\dim(U \cap W)$?
   1. 0
   2. 1
   3. 2
   4. 3
6. Question 6: Let U and W be subspaces of $\mathbb{R}^n$. If $\dim(U) + \dim(W) > n$, then what can you conclude about $U \cap W$?
   1. $U \cap W = {\vec{0}}$
   2. $U \cap W \neq {\vec{0}}$
   3. $\dim(U \cap W) = 0$
   4. $\dim(U \cap W) = n$
7. Question 7: If U is a subspace of W and $\dim(U) = 3$ and $\dim(W) = 5$, what is the dimension of $U \cap W$?
   1. 0
   2. 2
   3. 3
   4. 5