State the Definitions and use them in proofs.

• Right and left inverse of a matrix.
• Invertible matrix. Inverse of a matrix.
• Elementary matrix.
• LU decomposition of a matrix.
• Transpose of a matrix.
• Symmetric matrix; skew-symmetric matrix.
• Subspace of Rⁿ.
• Sum of subspaces of Rⁿ.
• Orthogonal complement of a subspace.
• Orthogonal subspaces.
• The four fundamental subspaces:
  • Null space
  • Column space
  • Row space
  • Left null space
• Sum of subspaces.
• Linearly independent set of vectors.
• Linearly dependent set of vectors.
• Basis of a vector space.
• Coordinates of a vector with respect to a basis.
• (Real) Vector space. Subspace of a vector space.
• Inner product in a vector space.

State the Theorems and use them in proofs.

• An m × n matrix A has
  • A right inverse if and only if rank A = m;
  • A left inverse if and only if rank A = n;
  • Both a right and a left inverse if and only if A is square (m=n) and of full rank; and then the left inverse and the right inverse are the same matrix.
• An n × n matrix is nonsingular if and only if it is invertible.
• If a square matrix has a one-sided inverse it is invertible.
• A product of invertible matrices is invertible; the inverse of the product is the product, in the reverse order, of the inverses of the factors.
• Elementary matrices are invertible; the inverse of an elementary matrix corresponds to the inverse row operation.
• If A is m × n, x ∈ Rⁿ and y ∈ R^m, then y ⋅ Ax = A^Ty ⋅ x.
• Subspaces
  • The span of a set of vectors is a subspace.
  • The solution set of a homogeneous system of linear equations is a subspace.
  • The sum of subspaces is a subspace.
  • The orthogonal complement of a set of vectors is a subspace.
  • Every subspace of Rⁿ other than the trivial subspace, has a basis.
  • If V is a subspace of Rⁿ, then (V ^⊥) ^⊥ =V.
  • Orthogonality relations among the four fundamental subspaces of a matrix.
• [Nonsingular matrix theorem.] The following are equivalent for an n × n matrix A:
  • A is nonsingular.
  • Ax=0 has only the trivial solution.
  • For every b in Rⁿ, the equation Ax=b has a solution (indeed, a unique solution).
  • A has a right inverse.
  • A has a left inverse.
  • The columns of A are a basis for Rⁿ.
• Appending a vector to a linearly independent set produces a new linearly independent set if and only if the new vector is not in the span of the original set.
• Two-out-of-three Theorem for bases: If S is a set of vectors in a k-dimensional vector space V, any two of the following imply the third.
  • S is a linearly independent set.
  • S spans V.
  • S has k elements.
• Theorem on bases for the four fundamental subspaces.
• Dimensions of the four fundamental subspaces.
• Rank-nullity Theorem.
• Dimension of the orthogonal complement of a subspace.
• Decomposition of a vector into components in, and orthogonal to, a subspace.

Algorithms: Do the following.

• Use row reduction to determine whether a square matrix is invertible and, if it is, to find its inverse.
• Use the shortcut to find the inverse of a 2 × 2 matrix.
• Interpret the dot product of vectors as the product of a one-row matrix with a one-column matrix.
• Use an outer product to find the standard matrix of a rank one projection.
• Given a matrix, determine a basis for each of its fundamental subspaces.

alex at iastate dot edu

Document last modified on Tue Oct 11 2011