MATH 284 - Linear Algebra

• Determine whether solutions of a linear system Ax = b exist, whether the solution is unique, and find a basis for the solution space.
• Explain what it means for a set of vectors to be a subspace of Rn and verify that a given set does or does not satisfy the defining properties of a subspace.
• Explain the concepts of linear independence, spanning, and basis.
• Determine whether a given set of vectors is linearly independent and/or spans a given subspace and produce a basis for a given subspace of Rn.
• Perform matrix calculations, applying the rules of matrix algebra.
• Find the column space, row space, and null space of a matrix and explain the relationship between the dimension of the null space, the rank, and the number of columns of the matrix.
• Define what it means for a function to be a linear transformation from Rn to Rm and describe the kernel and range of a given linear transformation.
• Produce the eigenvalues and associated eigenspaces for a given matrix and explain geometrically the result of multiplying an eigenvector by the matrix.
• Apply the dot product and its properties to problems of orthogonality, the magnitude of vectors, and the distance between vectors, and produce orthogonal bases of subspaces of Rn.
• Use the techniques and theory of linear algebra to model various real-world problems. 
• Effectively communicate the concepts and applications of linear algebra using the language of linear algebra in a mathematically correct way.
• Use advanced software tools (e.g., Maple, MATLAB, Mathematica) to solve problems in linear algebra.

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