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Linear Algebra II: Matrix Algebra
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Your ability to apply the concepts that we introduced in our previous course is enhanced when you can perform algebraic operations with matrices. At the start of this class, you will see how we can apply the Invertible Matrix Theorem to describe how a square matrix might be used to solve linear equations. This theorem is a fundamental role in linear algebra, as it synthesizes many of the concepts introduced in the first course into one succinct concept.
You will then explore theorems and algorithms that will allow you to apply linear algebra in ways that involve two or more matrices. You will examine partitioned matrices and matrix factorizations, which appear in most modern uses of linear algebra. You will also explore two applications of matrix algebra, to economics and to computer graphics.
Students taking this class are encouraged to first complete the first course in this series, linear equations.
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Upon completion of this course, learners will be able to:
- Apply matrix algebra, the matrix transpose, and the zero and identity matrices, to solve and analyze matrix equations.
- Apply the formal definition of an inverse, and its algebraic properties, to solve and analyze linear systems.
- Characterize the invertibility of a matrix using the Invertible Matrix Theorem.
- Apply partitioned matrices to solve problems regarding matrix invertibility and matrix multiplication.
- Compute an LU factorization of a matrix and apply the LU factorization to solve systems of equations.
- Apply matrix algebra and inverses to solve and analyze Leontif Input-Output problems.
- Construct transformation matrices to represent composite transforms in 2D and 3D using homogeneous coordinates.
- Construct a basis for a subspace.
- Calculate the coordinates of a vector in a given basis.
- Characterize a matrix using the concepts of rank, column space, and null space.
- Apply the Rank, Basis, and Matrix Invertibility theorems to describe matrices, subspaces, and systems.