Math 307 Final Exam Review

You will be asked for complete definitions (including appropriate context, e.g., what kind of thing might be linearly independent?) of some subset of the following terms. This includes all the terms from the first exam, plus new ones from the material covered since then. Remember that a complete mathematical definition leaves no room for ambiguity: if you give me a definition of what it means for a list of vectors to be linearly independent, I need to be able to use it, in principle anyway, to decide whether or not any list of vectors I ever meet is linearly independent.

• vector space
• subspace
• direct sum
• linear combination
• span
• finite dimensional
• linearly independent
• basis
• standard basis
• dimension
• linear map
• null space
• range
• injective
• surjective
• matrix of a linear map
• invertible
• isomorphism
• invariant subspace
• eigenvalue
• eigenvector
• upper triangular
• diagonalizable
• norm
• inner product
• orthogonal
• orthonormal
• orthogonal complement
• orthogonal projection
• linear functional
• adjoint
• self-adjoint
• normal
• positive semi-definite
• positive definite
• isometry
• polar decomposition
• singular values
• singular value decomposition
• change of basis matrix
• similar matrices
• unitary matrix
• orthogonal matrix
• generalized eigenvector
• multiplicity
• characteristic polynomial
• trace
• determinant
• cofactor

In lieu of review problems, I advise you primarily to study the homework problems. Do them again from scratch, compare your solutions to the ones I handed out in class, then do them again a few days later. For more preparation, start on other problems from the book.

Also, I suggest reading this article about study habits.

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