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Chapter 5: Problem 21

Suppose \(F: V \rightarrow U\) and \(G: U \rightarrow W\) are linear. Prove (a) \(\operatorname{rank}(G \circ F) \leq \operatorname{rank}(G)\) (b) \(\operatorname{rank}(G \circ F) \leq \operatorname{rank}(F)\)

Short Answer

Expert verified
To prove (a) rank(G 鈿� F) 鈮� rank(G), we show that Im(G 鈿� F) is a subspace of Im(G). Then, using the properties of subspaces and dimensions, the inequality rank(G 鈿� F) 鈮� rank(G) holds. To prove (b) rank(G 鈿� F) 鈮� rank(F), we create a linear transformation H that maps the pre-images of F to Im(F) and analyze their dimensions. Using the properties of linear transformations and dimensions, the inequality rank(G 鈿� F) 鈮� rank(F) holds.

Step by step solution

01

a) Proving rank(G 鈿� F) 鈮� rank(G)

1. Define the given linear transformations and their compositions: Here, we are given two linear transformations F: V 鈫� U and G: U 鈫� W, and we are analyzing the composition G 鈿� F: V 鈫� W, which means G(F(v)) for all v in V. 2. Identify the images of F and G 鈿� F: Let's denote Im(F) as the image of F, which is a subspace of U. Similarly, we will denote Im(G 鈿� F) as the image of G 鈿� F, which is a subspace of W. 3. Analyze the relationship between the images of G 鈿� F and G: Note that for each element in Im(G 鈿� F), we can write it as G(F(v)) for some v in V. Since G is a linear transformation, G(F(v)) is also an element in Im(G), which means that Im(G 鈿� F) is a subspace of Im(G). 4. Use the properties of subspaces and dimensions: Since Im(G 鈿� F) is a subspace of Im(G), we know that the dimension of Im(G 鈿� F) is less than or equal to the dimension of Im(G). In other words, rank(G 鈿� F) 鈮� rank(G).
02

b) Proving rank(G 鈿� F) 鈮� rank(F)

1. Define the given linear transformations and their compositions: Here, we are given two linear transformations F: V 鈫� U and G: U 鈫� W, and we are analyzing the composition G 鈿� F: V 鈫� W, which means G(F(v)) for all v in V. 2. Identify the images of F and G 鈿� F: Let's denote Im(F) as the image of F, which is a subspace of U. Similarly, we will denote Im(G 鈿� F) as the image of G 鈿� F, which is a subspace of W. 3. Analyze the relationship between the pre-images of F and G 鈿� F: The set of all vectors that map to Im(G 鈿� F) through F is a subset of the set of all vectors that map to Im(F) through F. In other words, the pre-image of Im(G 鈿� F) under F is a subset of the pre-image of Im(F) under F. 4. Create a function that maps the pre-images of F and analyzes the dimensions: Let's create a function H: V 鈫� U which maps the pre-image of Im(G 鈿� F) under F to Im(G 鈿� F) and the pre-image of Im(F) under F to Im(F). Since we've already established that the pre-image of Im(G 鈿� F) under F is a subset of the pre-image of Im(F) under F, H must be a linear transformation. 5. Use the properties of linear transformations and dimensions: Since H is a linear transformation, the dimension of the domain of H (pre-image of Im(G 鈿� F) under F) is less than or equal to the dimension of the codomain of H (pre-image of Im(F) under F). In other words, rank(G 鈿� F) 鈮� rank(F).

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