*Gidon Eshel*

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691128917
- eISBN:
- 9781400840632
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691128917.003.0003
- Subject:
- Environmental Science, Environmental Studies

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 ...
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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.Less

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.