## Course Content and Outcome Guide for MTH 261

- Course Number:
- MTH 261
- Course Title:
- Applied Linear Algebra I
- Credit Hours:
- 5
- Lecture Hours:
- 50
- Lecture/Lab Hours:
- 0
- Lab Hours:
- 0
- Special Fee:

#### Course Description

Surveys linear algebra with some applications. Includes linear systems, vectors, and vector spaces, including eigenspaces. Graphing calculator required. TI-89 Titanium or Casio Classpad 330 recommended. Prerequisites: MTH 253 and its prerequisite requirements. Audit available.#### Addendum to Course Description

This course is designed to familiarize students with the elementary concepts of linear algebra. The emphasis of the course is applications of linear algebra; abstract theory is kept to a minimum. Upon completion of the course, students will be familiar with the vocabulary of linear algebra and will have been exposed to numerous applications.

#### Intended Outcomes for the course

** ** Upon completion of this course the learner should be able to do the following things:

• Analyze real world scenarios to recognize when vectors, matrices, or linear systems are appropriate, formulate problems about the scenarios, creatively

model these scenarios (using technology, if appropriate) in order to solve the problems using multiple approaches, judge if the results are reasonable, and then interpret and clearly communicate the results.

• Appreciate linear algebra concepts that are encountered in the real world, understand and be able to communicate the underlying mathematics involved

to help another person gain insight into the situation.

• Work with vectors, matrices, or linear systems symbolically and geometrically in various situations and use correct mathematical terminology, notation,

and symbolic processes in order to engage in work, study, and conversation on topics involving vectors, matrices, or systems of linear equations with

colleagues in the field of mathematics, science or engineering.

#### Course Activities and Design

In-class activities are primarily lecture/discussion and problem-solving sessions. The students may use appropriate technology to investigate and reinforce concepts presented in class. A graphing calculator is required.

#### Outcome Assessment Strategies

1. Demonstrate an understanding of various types of linear systems and their applications to real world problems in:

#### Course Content (Themes, Concepts, Issues and Skills)

**SKILLS**

__Context Specific Skills__

**for at least 3**

*by hand***3 matrices.**

*x*__Learning Process Skills__

**THEMES CONCEPTS, and ISSUES**

**1.0**

**MATRIX ALGEBRA**

*m*-linear equations in

*n*-variables.

*mxn*matrix to reduced row echelon form using elementary row operations and Gauss-Jordan elimination.

*m*-linear equations in

*n*-variables by transforming its augmented matrix to reduced row echelon form.

*m*equations and

*n*unknowns to yield predetermined solutions in terms of geometries. For example, the solution to a 5 x 4 system should take on the form of a line off of the origin in

**R**

^{4}*mxn*matrices. Study the algebraic properties of

*mxn*matrices under these operations.

*mxn*matrix and demonstrate the ability to use transpose properties (sums, scalar products, products, transpose of a transpose, etc).

*mxn*matrix and an

*nxp*matrix. Study the algebraic properties of matrix multiplication.

**I**. Study the identity property of matrix multiplication.

**.**

**2.0**

**VECTOR SPACES**

*n*-Dimensional Euclidean Space,

**R**

^{n}.**R**. Study the algebraic properties of vectors under vector addition and scalar multiplication.

^{n}**R**. Solve a linear system to determine if a particular vector is a linear combination of a given set of vectors.

^{n}**R**. Determine the linear dependence or independence of a given set of vectors by solving a homogeneous linear system.

^{n}*mxn*matrix

**A**and study properties and the significance of these subspaces.

**.**Define the coordinates of a particular vector with respect to a given ordered basis. Compute the coordinates of a vector with respect to a given ordered basis. Use a transition matrix to convert the coordinates for a vector with respect to one ordered basis to its coordinates with respect to a second ordered basis.

**R**

^{n}. Study algebraic properties of the dot product.

**R**

^{n}.

**R**

^{n}with respect to orthogonal and orthonomal bases using the dot product.

**R**

^{n}into components lying in, and orthogonal to a given subspace of

**R**

^{n}. Construct the projection matrix relative to a given subspace.

**R**

^{n}using the Gram-Schmidt process.

**into a product QR, for Q an orthogonal matrix, and Q'R', for Q' an**

*A***orthonormal**matrix. In this case,

**R' = (Q')**

^{t}**A**

**3.0**

**LINEAR TRANSFORMATIONS**

*T: V -> W*. If N(

*T*) is the null space of

*T*, then N(

*T*) is a subspace of V. If R(

*T*) is the range of

*T*, then R(

*T*) is a subspace of W.

*T: V -> W*is a linear transformation, then dim(V) = dim(R(

*T*))+ dim(N(

*T*)).

*T:*

**R**

^{n}*->*

**R**

^{m}relative to the standard bases for

**R**

^{n}and

**R**

^{m}. Compute the matrix representation of such a linear transformation.

*T:*

**R**

^{n}*->*

**R**

^{m}relative to arbitrary ordered bases,

*B*

_{1}and

*B*

_{2}, for

**R**

^{n}and

**R**

^{m}, respectively. Compute the matrix representation of such a linear transformation relative to

*B*

_{1}and

*B*

_{2}.

*T: V -> W*, where

*V*and

*W*are arbitrary vector spaces, relative to ordered bases,

*B*

_{1}and

*B*

_{2}, for

*V*and

*W*, respectively. Compute the matrix representation of such a linear transformation relative to

*B*

_{1}and

*B*

_{2}.

*nxn*matrix,

**A**. Calculate eigenvalues by solving the characteristic equation of the matrix, det

**(A- Î»I)**=0. Calculate eigenvectors by solving the linear system

**(A- Î»I)x=0**.

**R**

^{n}. Define eigenvectors and eigenvalues of a linear operator,

*T*on a vector space

*V*.

*geometric*and

*algebraic*multiplicities for all eigenvalues of a given an

*n x n*matrix A. The

*geometric*multiplicity of Î» equals the nullity of (A- Î»I).

*nxn*matrix,

**A**. Define an eigenbasis for

**A**. Prove an

*nxn*matrix

**A**is diagonalizable if and only if

**R**

^{n}has an eigenbasis for

**A.**Study diagonalizable matrices in terms of the matrix representation of a linear transformation relative to an eigenbasis. Diagonalize a diagonalizable

*nxn*matrix.

**A = PDP**

^{-1}, then

**A**

^{n}=

**P D**

^{n}

**P**

^{-1}.

**4.0**

**APPLICATIONS**