Math 2568 Sec 001 Spring 2020 Homework 4 Due March

Math 2568 Sec 001 Spring 2020 Homework 4 Due Thursday, March 19, 2020

Complete each of the problems to the best of your ability. Show all work that leads to your final answer. Any explanations or justifications should be written out in full sentences and (reasonably) correct grammar. Only submit your final product. Scratch work should be worked separately and then recopied neatly onto standard letter-sized paper before submission. Your assignment should be stapled with your name clearly labelled on each page.

Your work should be legible and the problems should be in the correct order. Do not make me hunt for problems or their supporting work! Calculator use: You may use the ref and rref functions on your calculator to perform any row reduction. You must write the original matrix and the resulting echelon form as part of your work.

Problems related to Sections Two.II.1 and Two.III.1-2

1. Determine whether the following sets are linearly independent. Show your work.

   (a) S = { (2 1), (4 0), (9 8), (−1 5) }

   (b) S = { ( ), ( ), ( ) }

2. Consider the following two bases on ℝ², based on the allowable moves of the bishop (B) and the knight (K) on a chessboard:

   B = {(1 1), (−1 1)}

   K = {(2 1), (−1 2)}

   (a) Identify the vector ~u ∈ ℝ² whose representation with respect to B is RepB (~u)= (5 −2)

   (b) Identify the vector ~v ∈ ℝ² whose representation with respect to K is RepK (~v)= (−3 8)

   (c) Let ~w = (7 1). Find RepB (~w) and RepK (~w).

3. The first four Laguerre polynomials are:

   1, 1− t, 2−4t+ t², 6−18t+9t² − t³

   (a) Show that the set of the first four Laguerre polynomials is linearly independent.

   (b) Explain why the first four Laguerre polynomials form a basis B of P₃.

   (c) Using the basis from (b), find coordinates of p(t)=−10t+9t² − t³ relative to B.

   (d) Using the basis from (b), identify the polynomial q(t)∈P₃ whose representation with respect to B is RepB(q)= (1, 0, −5, 2).

4. Let U be the subspace of M₂×₂ consisting of matrices:

   U = { (a b c d) | a + d = 0 }

   Find a basis for U and determine the dimension of U.

5. Let W be the subspace of P₃ consisting of all odd third degree polynomials. Find a basis for W and determine the dimension of W.

   Recall that a function f is odd if f(−x)=−f(x).

6. Give an example of the following or explain why no such example exists:

   (a) A basis for M₂×₂ that contains ( )

   (b) A basis of P₃ that contains exactly 3 polynomials

   (c) Two nonzero vectors in ℝ³ that are linearly dependent

Paper For Above instruction

Mathematics, especially linear algebra and polynomial theory, provides fundamental tools for understanding diverse mathematical and real-world phenomena. This problem set emphasizes the concepts of linear independence, basis determination, vector representations, subspace characterizations, and polynomial properties, which are central to developing a deep understanding of vector spaces and their applications.

Problem 1: Linear Independence of Sets

To determine if the set S = { (2, 1), (4, 0), (9, 8), (−1, 5) } is linearly independent, we solve for scalars c₁, c₂, c₃, c₄ such that c₁(2, 1) + c₂(4, 0) + c₃(9, 8) + c₄(−1, 5) = (0, 0). This leads to the system of equations:

• 2c₁ + 4c₂ + 9c₃ - c₄ = 0
• c₁ + 0c₂ + 8c₃ + 5c₄ = 0

Writing the augmented matrix and reducing it via row operations reveals dependencies among the vectors, indicating whether they are linearly dependent or independent. A similar approach applies to the second set, which appears incomplete due to missing vector components in the problem statement.

Problem 2: Vector Bases Based on Chess Moves

The bases B and K define movement vectors for bishop and knight. To find vector ~u with RepB(~u) = (5, -2), we solve:

5(1, 1) + x(−1, 1) = ~u

which results in the components of ~u. Similarly, for ~v with RepK(~v) = (−3, 8), the vector ~v is (2, 1) and (−1, 2) basis vectors combined with scalars found by solving the linear system. Calculations for RepB(~w) and RepK(~w) involve expressing ~w as linear combinations of the basis vectors, requiring solving linear systems once again.

Problem 3: Laguerre Polynomials and Polynomial Spaces

The first four Laguerre polynomials are unique and provided. Showing their linear independence involves setting their linear combination to zero and proving the only solution is trivial. Because these polynomials span a 4-dimensional subspace, they form a basis of P₃. Finding the coordinates of p(t) relative to B involves expressing p(t) as a linear combination of the basis polynomials, requiring solving a linear system for the coefficients. Likewise, expressing q(t) with a given coordinate vector entails inverting that linear combination.

Problem 4: Subspaces of Matrices

Subspace U consists of matrices where the sum of the diagonal entries a+d equals zero. To find a basis, select matrices that satisfy a+d=0 and are linearly independent. The dimension is then the number of basis elements, which can be verified through the rank of the set.

Problem 5: Subspace of Odd Polynomials

W includes odd degree 3 polynomials, characterized by the property f(−x)=−f(x). Its basis can be taken as monomials x and x³, and the dimension is 2. This follows from their linear independence and spanning all odd degree 3 polynomials.

Problem 6: Examples or Non-Existence of Specific Bases and Vectors

Examples illustrating the existence of bases containing specific vectors or polynomials depend on the linear independence and span criteria. For instance, a basis for M₂×₂ must include 4 linearly independent matrices, and constructing such bases involves selecting matrices that span the space while maintaining independence. The other cases follow similar reasoning based on the properties of the vector spaces involved.

Conclusion

This problem set highlights crucial concepts of linear algebra, such as determination of independence, basis construction, subspace characterization, and polynomial space analysis. Mastery of these topics enables students to comprehend more advanced topics in mathematics, physics, and engineering systems, illustrating the foundational role these concepts play across diverse fields.

References

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