The operations used to perform row reduction are called row operations.

We summarize the notation to keep track of the precise row operations being used.

We summarize the algorithm for performing row reduction.

Matrix algebra uses three different types of operations.

Matrices and vectors can be used to rewrite systems of equations as a single equation, and there are advantages to doing this.

Sums of solution to homogeneous systems are also solutions.

Row and column operations can be performed using matrix multiplication.

We begin our introduction to vector spaces with the concrete example of $\mathbb {R}^n$.

A vector space is a set equipped with two operations, vector addition and scalar multiplication, satisfying certain properties.

A subset of a vector space is a subspace if it is non-empty and, using the restriction to the subset of the sum and scalar product operations, the subset satisfies the axioms of a vector space.

A linear combination is a sum of scalar multiples of vectors.

A collection of vectors spans a set if every vector in the set can be expressed as a linear combination of the vectors in the collection. The set of rows or columns of a matrix are spanning sets for the row and column space of the matrix.

Nullspaces provide an important way of constructing subspaces of $\mathbb R^n$.

Another subspace associated to a matrix is its range.

A basis is a collection of vectors which consists of enough vectors to span the space, but few enough vectors that they remain linearly independent. It is the same as a minimal spanning set.

An array of numbers can be used to represent an element of a vector space.

To complete this section we extend our set of scalars from real numbers to complex numbers.

A linear transformation is a function between vector spaces preserving the structure of the vector spaces.

A linear transformation can be represented in terms of multiplication by a matrix.

Determine how the matrix representation depends on a choice of basis.

The collection of all linear transformations between given vector spaces itself forms a vector space.

One method for computing the determinant is called cofactor expansion.

There is also a combinatorial approach to the computation of the determinant.

The determinant is connected to many of the key ideas in linear algebra.

A nonzero vector which is scaled by a linear transformation is an eigenvector for that transformation.

The span of the eigenvectors associated with a fixed eigenvalue define the eigenspace corresponding to that eigenvalue.

Establish algebraic criteria for determining exactly when a real number can occur as an eigenvalue of $A$.

The subspace spanned by the eigenvectors of a matrix, or a linear transformation, can be expressed as a direct sum of eigenspaces.

Similarity represents an important equivalence relation on the vector space of square matrices of a given dimension.

There are advantages to working with complex numbers.

There are advantages to working with complex numbers.

There are advantages to working with complex numbers.

There are advantages to working with complex numbers.

For an $n\times n$ complex matrix $A$, $\mathbb C^n$ does not necessarily have a basis consisting of eigenvectors of $A$. But it will always have a basis consisting of generalized eigenvectors of $A$.

The singular value decomposition is a genearlization of Shur’s identity for normal matrices.