If S is an isometry, then |det S| = 1. If T is an operator on R^n, then T changes volume by a factor of |det T|. This result leads to the change of varibales formula in integration in R^n....

From Sheldon Axler

Trace of an operator defined to be the sum of the eigenvalues (or of the eigenvalues of the complexification), repeated according to multiplicity. Trace of a matrix defined to be the sum of the squares of the diagonal enties. The connection between these ...

From Sheldon Axler

Description of isometries on real inner product spaces as direct sums of the identity operator, the negative of the identity operator, and rotations on subspaces of dimension 2....

From Sheldon Axler

Every operator on a finite-dimensional complex vector space has a matrix (with respect to some basis of the vector space) that is a block diagonal matrix, with each block itself an upper-triangular matrix that contains only one eigenvalue on the diagonal....

From Sheldon Axler

Decomposition of an operator on a complex vector space. For each operator on a complex vector space, there is a basis of the vector space consisting of generalized eigenvectors of the operator. Multiplicity (also called algebraic multiplicity) and geometr...

From Sheldon Axler

The analogy between the complex numbers and L(V). The Polar Decomposition: If T is an operator on a finite-dimensional inner product space V, then there exists an isometry on V such that T equals S times the square root of T*T....

From Sheldon Axler

Self-adjoint operators. All eigenvalues of a self-adjoint operator are real. On a complex vector space, if the inner product of Tv and v is real for every vector v, then T is self-adjoint....

From Sheldon Axler

Polynomials applied to an operator. Proof that every operator on a finite-dimensional, nonzero, complex vector space has an eigenvalue (without using determinants!)....

From Sheldon Axler

Linear functionals, dual spaces, dual bases, and dual maps....

From Sheldon Axler

Definitions of null space, injectivity, range, and surjectivity. Fundamental theorem of linear maps. Consequences for systems of linear equations....

From Sheldon Axler

Use the result that all bases of a vector space have the same length to define the dimension of the vector space. Show that every linearly independent list of the right length is a basis. Also, every spanning list of the right length is a basis. The formu...

From Sheldon Axler

Span, linear independence, the Linear Dependence Lemma, and the inequality between the length of a linearly independent list and the length of a spanning list....

From Sheldon Axler