Math 2568 Autumn 2020 Homework 4 For Each Of The Following

Math 2568 Autumn 2020 Homework 4 for Each Of The Following Sets Of Vecto

Find a minimal spanning set T for the subspace W = Span(S) from the set of vectors in S (T ⊆ S). For each vector v_i ∈ S omitted from T, write it explicitly as a linear combination of the vectors in T.

Paper For Above instruction

The following analysis addresses the task of determining minimal spanning sets for various collections of vectors and expressing omitted vectors as linear combinations of the spanning set. This process is fundamental in linear algebra for understanding the structure of subspaces within a vector space, such as R^n, and aids in simplifying the basis while preserving the span.

Problem 1:

Given the set S = {v₁, v₂, v₃, v₄, v₅, v₆} with respective vectors:

v₁ = [2, 3, -1, 4, 6],

v₂ = [9, -4, -1, 0, 0],

v₃ = [-30, 25, -47, -16, -2],

v₄ = [0, -1, 0, 0, 0],

v₅ = [-7, 1, 0, 0, 0],

v₆ = [35, -6, 11, -50, -34].

The goal is to find a minimal spanning set T ⊆ S for the subspace W = Span(S). We start by identifying linearly independent vectors among these, possibly by constructing a matrix with these vectors as rows or columns and reducing it to row echelon form.

Subsequently, vectors omitted from T should be expressed as linear combinations of vectors in T. For example, if v₁ is omitted, its coordinates should be written as a linear combination of the vectors in T, which are linearly independent and span W.

Problem 2:

Set S = {v₁, v₂, v₃, v₄} with vectors:

v₁ = [1, -1, 0, 0],

v₂ = [1, -1, 0, 0],

v₃ = [1, -1, 0, 0],

v₄ = [1, -1, 0, 0].

This set shows repeated vectors; thus, the minimal spanning set T will be a single vector, as all vectors are scalar multiples or identical. The span is therefore one-dimensional, with T = {v₁}. Vectors v₂, v₃, v₄ are linear combinations of v₁, specifically, each is equal to v₁.

Problem 3:

Set S = {v₁, v₂, v₃, v₄, v₅} with vectors:

v₁ = [2, 3, -1, 9, 5],

v₂ = [6, 4, 5, -3, 7],

v₃ = [-2, 2, -7, 21, 3],

v₄ = [1, 2, -1, 0, 0],

v₅ = [9, 6, 3, -2, 1].

Formally, determine a minimal set T that spans W and find linear combinations for vectors not in T. The process involves reducing the matrix formed by these vectors and identifying pivot columns to find the basis.

Problem 4:

Set S = {v₁, v₂, v₃, v₄} with:

v₁ = [1, 2, -1, 3],

v₂ = [-2, 1, 2, -1],

v₃ = [-1, -1, 1, -3],

v₄ = [-1, 0, 5, -14].

Identify the minimal spanning set T for the subspace spanned by S. Also, express vectors omitted from T as linear combinations of T, which involves solving systems of linear equations resulting from the matrix of these vectors.

Problems 5-8:

Given specific matrices, find minimal spanning sets for their column space, row space, and nullspace by performing suitable reductions and basis extractions.

Problems 9-10:

The first involves the concept of span and linear dependence, determining if the given statements are true. The second involves properties of matrices satisfying A * A = 0, and whether their column space is contained in their nullspace, which requires understanding linear transformations and subspace relations.

Summary

This collection of problems emphasizes the fundamental procedures in linear algebra for determining bases, minimal spanning sets, and expressing vectors as combinations of basis vectors. These concepts underpin the structure of vector spaces, subspaces, and linear transformations and are essential for advanced studies and applications in engineering, computer science, and mathematics.

References

  • Larry C. Wang, “Linear Algebra and Its Applications,” Pearson Education, 2017.
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