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Rank and mullity
Rank and nullity
Removing a linearly dependent vector from a set does not change the span of the set.
R^n is a vector space.
Row equivalence is an equivalence relation
Row equivalent matrices have the same row space.
Row equivalent matrices represent equivalent linear systems
Row operations
Row operations are given by multiplication by elementary matrices.
Row operations do not necessarily preserve the column space.
Scalar multiplication
Similarity of matrices
Similarity of matrices in an equivalence relation.
Similar matrices have the same eigenvalues and the same characteristic polynomials.
Spans
Subspaces
Subspaces associated to a linear transformation
Subspaces associated to a matrix
Switching two rows multiplies the determinant by -1.
Symmetric matrices
Symmetric matrices are square.
Terminology
The 0 scalar multiplied by any vector equals the 0 vector.
The 0 vector is unique.
The 0 vector multiplied by any scalar equals the 0 vector.

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