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# MTH208 Advanced Linear Algebra SUSS Assignment Sample

MTH208 Advanced Linear Algebra is an exciting course for students interested in deepening their understanding of linear algebra principles. In this class, you can expect to learn essential skills such as adding, subtracting and multiplying matrices, as well as solving systems of linear equations. Beyond the fundamentals, you will also gain hands-on experience applying concepts to real-world contexts and engaging in problem-solving activities. For those looking to expand their knowledge of mathematics beyond the basics, MTH208 provides the perfect opportunity!

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### Assignment Activity 1: Determine whether two given square matrices are similar.

Determining whether two given square matrices are similar is a complex process that involves conceptually understanding their properties. To begin, an assessment of their dimensions must start first to see if they both have the same size and shape. Then, eigenvalues must be extracted from both matrices as they house important information regarding their similarity. Lastly, after examining both matrices and comparing the eigenvalues between them, one can decide whether the two matrices being assessed are similar or not. This comparison is a key factor in determining the matrices’ resemblance to each other, providing an effective way to make this assessment.

### Assignment Activity 2: Demonstrate properties of normal operators.

Normal operators are indispensable in many different fields, ranging from physics and engineering to economics. Their properties make them incredibly powerful tools when it comes to solving linear equations with multiple variables, which makes them a valuable asset to anyone seeking an effective solution. Mathematically speaking, normal operators are characterized by having their eigenvectors close by orthogonal and with the same length.

They also have the unique ability to make known the coefficients of their variables when used in the adjoint form. Their spectrum is either compact or discrete, meaning its elements join together naturally and will not break that connection once formed. These attributes demonstrate why it is so important to study and understand normal operators, allowing us to apply them accurately within our field of choice.

### Assignment Activity 3: Employ properties of positive definite matrices.

Positive-definite matrices have useful properties that can be employed to great benefit. In particular, a positive definite matrix is always invertible; this means its inverse exists and can be used to solve systems of linear equations. Additionally, the Cholesky decomposition of a positive definite matrix yields two triangular matrices – a lower and an upper. This can be employed to help ensure numerical stability when solving numerical problems. Thus, understanding the properties of positive definite matrices is essential when dealing with many fields, from engineering to economics and finance.

### Assignment Activity 4: Calculate Jordan’s canonical form of a given square matrix with complex entries over the complex numbers.

The complex Jordan Canonical Form of a given square matrix with complex entries is an important concept in linear algebra, and can be calculated through a number of steps. First, the matrix must be reduced to a semi-diagonal form by applying similarity transformations to it; then, its eigenvectors and corresponding eigenvalues must be determined; finally, the matrix is divided into distinct non-zero blocks classified according to their eigenvalues. Each one of these steps requires careful analysis and understanding of the mathematics underlying them in order to guarantee a successful outcome. However, having done all that work, the results provide insightful information about the considered problem.

### Assignment Activity 5: Compute matrix representation of a given linear operator with respect to a fixed basis or the change of basis matrix from one basis to another basis.

Working in linear-algebraic spaces, the representation of a linear operator with respect to a fixed basis is concretely calculated by constructing its matrix form. By analyzing and expressing the action of this linear operator between two given sets, we are able to explicitly compute its representation as a matrix – especially when there is a set of fixed bases with specific members. The information required in this computation is obtained through an evaluation of the action of that linear operator on the basis required; once all evaluations are taken into account, we are provided with the necessary information to construct what can be referred to as a change of basis matrix: allowing us to transform one basis to another.

### Assignment Activity 6: Show how to prove a mathematical statement in linear algebra.

Proving a mathematical statement in linear algebra is all about finding the right strategy. For example, theorems can be helpful for one type of proof; which specify the conditions under which a statement must hold true. Equivalence proofs are another option when examining similarities between two expressions by using certain properties. Direct proofs meanwhile focus on a single statement and its logic.

As for indirect proofs, this is where you prove your statement by assuming it’s false and reaching a contradiction using other principles or axioms of linear algebra. Whichever path you choose, taking the time to understand linear algebra and its fundamentals will be beneficial in helping you determine which specific proof methods would be most fitting for each situation and statement.

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