# Matrix Properties, Fundamental Spaces, Orthogonality

# Matrix Properties, Fundamental Spaces, Orthogonality

This chapter provides an introduction to linear algebra. Topics covered include vector spaces, matrix rank, fundamental spaces associated with **A** ɛ ℝM×N, and Gram–Schmidt orthogonalization. In summary, an *M* × *N* matrix is associated with four fundamental spaces. The *column space* is the set of all *M*-vectors that are linear combinations of the columns. If the matrix has *M* independent columns, then the column space is ℝM; otherwise the column space is a subspace of ℝM. Also in ℝM is the left *null space*, the set of all *M*-vectors that the matrix’s s transpose maps to the zero *N*-vector. The *row space* is the set of all *N*-vectors that are linear combinations of the rows. If the matrix has *N* independent rows, then the row space is ℝN; otherwise, the row space is a subspace of ℝN. Also in ℝN is the *null space*, the set of all *N*-vectors that the matrix maps to the zero *M*-vector.

*Keywords:*
linear algebra, matrix, Gram–Schmidt orthogonalization, column space, row space, null space, vectors

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