Linear Algebra Assignment 2 2014 Due October 6

Sheet11mat1163 Linear Algebratextassignment 2 2014due 6 October 2014

Consider the matrix (a) Calculate the rref of the augmented matrix [A I] where I denotes the 3 by 3 identity matrix. (b) Is A invertible? (c) If your answer to (b) is yes, write down the inverse of A. (d) For each of the following statements decide if it is true or false: (i) A has two pivot positions. (ii) The equation Ax = b has at least one solution for each b in R3. (iii) The columns of A are linearly independent. (iv) The linear transformation x → Ax is one-to-one.

The set of vectors is not a subspace of R2. Prove this statement by:

1. Providing an example of a vector x and a scalar α such that x ∈ S but αx ∉ S.
2. Providing an example of vectors x and y such that x ∈ S and y ∈ S, but x + y ∉ S.

Consider the subspaces S1 and S2 defined by certain equations:

• (a) Identify which subspace the vector (1,3,1) belongs to: S1 or S2.
• (b) Determine a basis for that subspace.
• (c) Write the augmented matrix of the system of equations to find the coordinates of (1,3,1) relative to the basis, and compute these coordinates.
• (d) Describe geometrically the set of all vectors belonging to both S1 and S2.

Finite-dimensional matrices are examined through characteristic polynomial, eigenvalues, eigenvectors, and diagonalizability:

• (a) Determine the characteristic polynomial of A.
• (b) Find the eigenvalues of A.
• (c) For each eigenvalue, find a basis for its eigenspace.
• (d) Decide if A is diagonalizable.
• (e) If diagonalizable, find matrices P and D such that PD=AP.

Vectors u in R^2 whose components satisfy certain relations are analyzed:

• (a) For a given vector u, define P and Q, compute images Px and Qx, and find eigenvalues and eigenvectors of P and Q.
• (b) Repeat the process for a different vector u, defining new P and Q accordingly.
• (c) Describe relationships between eigenvalues and eigenvectors of P and Q, completing the sentences about their correspondence.

Paper For Above instruction

Introduction

Linear algebra forms the foundation of many mathematical and applied disciplines, including computer science, engineering, and physics. It deals with vector spaces, transformation properties, and matrix operations, which are essential tools for solving systems of linear equations, analyzing geometric structures, and understanding the behavior of linear transformations. This paper addresses a comprehensive set of problems related to the properties of matrices, subspaces, eigenvalues, eigenvectors, and linear transformations, engaging with both theoretical aspects and practical computation techniques.

Analysis of Matrices and Invertibility

The first problem examines matrix A, requiring calculation of the reduced row echelon form (rref) of the augmented matrix [A | I], where I is the 3×3 identity matrix. Computing the rref helps determine the invertibility of A; if the matrix reduces to the identity, A is invertible, and its inverse can be explicitly written. Conversely, if the matrix does not reduce to the identity, A is singular and not invertible. The calculation of the rref involves Gaussian elimination, transforming the augmented matrix to a form that clearly shows the pivot positions and the solutions to the system.

Determining whether A is invertible depends on whether the matrix has three pivots, which indicates full rank. If invertible, the inverse matrix can be found via the Gauss-Jordan elimination process. Based on the properties of invertible matrices, certain statements about the positions of pivots, solutions to the equation Ax=b for all b in R3, column independence, and the injectivity of the transformation x → Ax are evaluated as true or false. These properties are interconnected, as invertibility implies the uniqueness of solutions and linear independence of columns, which correspond to the matrix's rank and the transformation being one-to-one.

Subspace Verification and Vector Examples

The second problem assesses the subspace status of a set of vectors in R2. To prove that a set is not a subspace, it is sufficient to find a vector x within the set and a scalar α such that αx does not belong to the set, violating the closure under scalar multiplication. Similarly, demonstrating non-closure under vector addition involves finding vectors x and y both in the set but whose sum is not in the set. Specific examples are constructed following the defining properties of the set, illustrating the failure of subspace criteria. These counterexamples highlight the importance of the subspace axioms—closure under addition and scalar multiplication—in linear algebra.

Subspace Subsets and Geometric Interpretation

Problems involving subspaces S1 and S2 specified by linear equations in R3 involve identification of the subspace to which a particular vector belongs. Using systems of equations, the basis of the relevant subspace is found by selecting vectors that satisfy the defining equations. The augmented matrix approach aids in solving for coordinates of vectors relative to the basis, providing a change-of-basis perspective essential for understanding linear combinations and span.

The set of vectors common to both subspaces (the intersection S1 ∩ S2) is interpreted geometrically as a line or a plane in R3, depending on the constraints. The vector equation describing this intersection involves parametric representations—linear combinations of basis vectors scaled by parameters—while the geometric description offers insight into the dimensionality and structure of the shared subspace.

Eigenvalues, Eigenvectors, and Diagonalization

The fourth problem explores the spectral properties of matrix A through the characteristic polynomial and eigenvalues. Eigenvalues are roots of the characteristic polynomial, and the associated eigenvectors are solutions to the homogeneous system (A - λI)x = 0. Determining eigenspaces involves finding bases for these solution spaces. If A has a complete set of eigenvectors, it is diagonalizable, which simplifies many matrix computations. When diagonalizable, a matrix P composed of eigenvectors and a diagonal matrix D with eigenvalues satisfy PD=AP.

Analysis of Vector Transformations and Eigen Concepts

The final problems involve vectors u in R2, with components satisfying specific relations, and linear transformations P and Q defined by these vectors. Calculations include applying these transformations to specific vectors, deriving new vectors Px and Qx, and analyzing their properties such as eigenvalues and eigenvectors. The relations between the eigenvalues and eigenvectors of P and Q are explored, illustrating the correspondence between spectral properties under different transformations.

This comprehensive analysis emphasizes the interconnectedness of properties like invertibility, linear independence, spectral characteristics, and geometric interpretations in the study of linear algebra. Mastery of these concepts allows for solving diverse problems, from theoretical proofs to practical computations.

Conclusion

The problems presented showcase the richness of linear algebra, highlighting the importance of matrix operations, subspace properties, eigenvalues, and eigenvectors. These tools underpin many advanced topics and applications across scientific disciplines. Understanding the detailed computations and theoretical underpinnings equips students and practitioners with a robust foundation for further exploration and problem-solving in linear algebra.

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