18.03 Lecture Notes - Lecture 12: Differential Operator, Augmented Matrix, Gaussian Elimination

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Be able to draw conclusions about solutions to a system from its augmented matrix in row-echelon form. Understand the importance of and be able to nd the four fundamental subspaces of a matrix. Understand the relationship between the four fundamental subspaces. Understand the relationship between the number of columns, rank, and nullity of a matrix. Find a basis for the vector space spanned by a set of vectors. Understand how a matrix can be thought of as a linear transformation. Understand how to write a linear transformation in matrix form. Consider a linear system of m equations in n variables. Let b be the m x (n + 1) augmented matrix after being put in row-echelon form. Case 1: the matrix b has a pivot in the augmented column. This means that one of the new equations has the form. Case 2: the matrix b has no pivot in the augmented column.

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