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Let P: R4 a) b) - R4 be given by P((x1, x2, x3, x4)) = (x₁,0, x3,0). Prove that P is a linear map. Find the matrix of P with respect to the canonical basis of R4. Find a basis for Nul (P) and dim(Nul(P)).

Let P: R4 a) b) - R4 be given by P((x1, x2, x3, x4)) = (x₁,0, x3,0). Prove that P is a linear map. Find the matrix of P with respect to the canonical basis of R4. Find a basis for Nul (P) and dim(Nul(P)).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please solve problem 5
### Question 5: Let \(P : \mathbb{R}^4 \rightarrow \mathbb{R}^4\) be given by \[P((x_1, x_2, x_3, x_4)) = (x_1, 0, x_3, 0).\] a) **Prove that \(P\) is a linear map.** b) **Find the matrix of \(P\) with respect to the canonical basis of \(\mathbb{R}^4\).** c) **Find a basis for \( \operatorname{Nul}(P) \) and \( \dim(\operatorname{Nul}(P)) \).**
Transcribed Image Text:### Question 5: Let \(P : \mathbb{R}^4 \rightarrow \mathbb{R}^4\) be given by \[P((x_1, x_2, x_3, x_4)) = (x_1, 0, x_3, 0).\] a) **Prove that \(P\) is a linear map.** b) **Find the matrix of \(P\) with respect to the canonical basis of \(\mathbb{R}^4\).** c) **Find a basis for \( \operatorname{Nul}(P) \) and \( \dim(\operatorname{Nul}(P)) \).**
### Linear Algebra Exercises #### Problem d) - **Task**: Find a basis for \( \text{Ran}(P) \) and \( \dim(\text{Ran}(P)) \). #### Problem e) - **Task**: Find \( P \circ P \). #### Problem f) - **Task**: Prove that \( \mathbb{R}^4 = \text{Nul}(P) \oplus \text{Ran}(P) \). --- ### Explanation #### Problem d) The range of a linear transformation \( P \), denoted as \( \text{Ran}(P) \), is the set of all possible outputs of \( P \). Finding a basis for \( \text{Ran}(P) \) involves determining a set of linearly independent vectors that span the range. The dimension of the range, \( \dim(\text{Ran}(P)) \), is the number of vectors in this basis. #### Problem e) The composition of \( P \) with itself, \( P \circ P \), or \( P^2 \), is defined by applying \( P \) twice to any vector. If \( P: V \rightarrow V \) is a linear transformation on a vector space \( V \), then \( P \circ P: V \rightarrow V \). #### Problem f) To prove \( \mathbb{R}^4 = \text{Nul}(P) \oplus \text{Ran}(P) \), we must show that every vector in \( \mathbb{R}^4 \) can be uniquely expressed as a sum of a vector in the null space of \( P \) and a vector in the range of \( P \). The direct sum \( \oplus \) indicates that these subspaces intersect trivially (only at the zero vector). This decomposition highlights the fundamental theorem of linear algebra for finite-dimensional vector spaces, linking the concepts of kernel (null space) and image (range).
Transcribed Image Text:### Linear Algebra Exercises #### Problem d) - **Task**: Find a basis for \( \text{Ran}(P) \) and \( \dim(\text{Ran}(P)) \). #### Problem e) - **Task**: Find \( P \circ P \). #### Problem f) - **Task**: Prove that \( \mathbb{R}^4 = \text{Nul}(P) \oplus \text{Ran}(P) \). --- ### Explanation #### Problem d) The range of a linear transformation \( P \), denoted as \( \text{Ran}(P) \), is the set of all possible outputs of \( P \). Finding a basis for \( \text{Ran}(P) \) involves determining a set of linearly independent vectors that span the range. The dimension of the range, \( \dim(\text{Ran}(P)) \), is the number of vectors in this basis. #### Problem e) The composition of \( P \) with itself, \( P \circ P \), or \( P^2 \), is defined by applying \( P \) twice to any vector. If \( P: V \rightarrow V \) is a linear transformation on a vector space \( V \), then \( P \circ P: V \rightarrow V \). #### Problem f) To prove \( \mathbb{R}^4 = \text{Nul}(P) \oplus \text{Ran}(P) \), we must show that every vector in \( \mathbb{R}^4 \) can be uniquely expressed as a sum of a vector in the null space of \( P \) and a vector in the range of \( P \). The direct sum \( \oplus \) indicates that these subspaces intersect trivially (only at the zero vector). This decomposition highlights the fundamental theorem of linear algebra for finite-dimensional vector spaces, linking the concepts of kernel (null space) and image (range).
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