A Summary of Linear Algebra
John Mitchell

This document is a list of some material in linear algebra that you should be familiar with. Throughout, we will take A to be the 3 x 4 matrix I assume you are familiar with matrix and vector addition and multiplication.

• All vectors will be column vectors.
• Given a vector v, if we say that , we mean that v has at least one nonzero component.
• The transpose of a vector or matrix is denoted by a superscript T. For example, • The inner product or dot product of two vectors u and v in can be written uTv; this denotes . If uTv=0 then u and v are orthogonal.
• The null space of A is the set of all solutions x to the matrix-vector equation Ax=0.
• To solve a system of equations Ax=b, use Gaussian elimination. For example, if , then we solve Ax=b as follows: (We set up the augmented matrix and row reduce (or pivot) to upper triangular form.) Thus, the solutions are all vectors x of the form for any numbers s and t.
• The span of a set of vectors is the set of all linear combinations of the vectors. For example, if and then the span of v1 and v2 is the set of all vectors of the form sv1+tv2 for some scalars s and t.
• The span of a set of vectors in gives a subspace of . Any nontrivial subspace can be written as the span of any one of uncountably many sets of vectors.
• A set of vectors is linearly independent if the only solution to the vector equation is for all i. If a set of vectors is not linearly independent, then it is linearly dependent. For example, the rows of A are not linearly independent, since To determine whether a set of vectors is linearly independent, write the vectors as columns of a matrix C, say, and solve Cx=0. If there are any nontrivial solutions then the vectors are linearly dependent; otherwise, they are linearly independent.
• If a linearly independent set of vectors spans a subspace then the vectors form a basis for that subspace. For example, v1 and v2 form a basis for the span of the rows of A. Given a subspace S, every basis of S contains the same number of vectors; this number is the dimension of the subspace. To find a basis for the span of a set of vectors, write the vectors as rows of a matrix and then row reduce the matrix.
• The span of the rows of a matrix is called the row space of the matrix. The dimension of the row space is the rank of the matrix.
• The span of the columns of a matrix is called the range or the column space of the matrix. The row space and the column space always have the same dimension.
• If M is an m x n matrix then the null space and the row space of M are subspaces of and the range of M is a subspace of .
• If u is in the row space of a matrix M and v is in the null space of M then the vectors are orthogonal. The dimension of the null space of a matrix is the nullity of the matrix. If M has n columns then rank(M)+nullity(M)=n. Any basis for the row space together with any basis for the null space gives a basis for .
• If M is a square matrix, is a scalar, and x is a vector satisfying then x is an eigenvector of M with corresponding eigenvalue . For example, the vector is an eigenvector of the matrix with eigenvalue .
• The eigenvalues of a symmetric matrix are always real. A nonsymmetric matrix may have complex eigenvalues.
• Given a symmetric matrix M, the following are equivalent:
1.
All the eigenvalues of M are positive.
2.
xTMx>0 for any .
3.
M is positive definite.
• Given a symmetric matrix M, the following are equivalent:
1.
All the eigenvalues of M are nonnegative.
2. for any x.
3.
M is positive semidefinite.

John E. Mitchell
2004-08-31