Chapter 1: Vector Spaces

Additive Inverse

A vector that adds to a given vector to produce the zero vector.

Basis

A set of vectors that is linearly independent and spans an entire vector space.

Cancellation Law

If the same vector is added to two vectors and the results are equal, then the original two vectors are equal.

Closed Under Addition

Adding any two vectors in a set always gives another vector in that same set.

Closed Under Scalar Multiplication

Multiplying any vector in a set by any scalar always gives another vector in that same set.

Finite-Dimensional Space

A vector space that has a basis containing finitely many vectors.

Generates

A set generates a vector space if every vector in the space can be made from a linear combination of vectors in the set

Infinite-Dimensional Space

A vector space that has no finite basis

Linear Combination

A vector formed by multiplying vectors by scalars and then adding the results.

Linearly Dependent

A set of vectors is when at least one vector can be made from a linear combination of the others.

Linearly Independent

A set of vectors is when none of its vectors can be made from a linear combination of the others

Maximal Linearly Independent

A linearly independent set that cannot have another vector from the space added to it while remaining linearly independent.

Scalar

A number from the field used for multiplying vectors.

Scalar Multiplication

The operation of multiplying a vector by a scalar.

Span of a Subset

The collection of every linear combination that can be formed from vectors in the subset.

Spans

A set of vectors spans a space when every vector in the space is a linear combination of vectors from that set.

Subspace

A subset of a vector space that is itself a vector space under the same addition and scalar multiplication operations.

Subspace Generated by Elements of a Set

The smallest subspace containing all elements of that set; equivalently, all linear combinations of its elements.

Trivial Representation of Zero

A representation of the zero vector as a linear combination in which every scalar coefficient is zero.

Vector

An object in a vector space that can be added to other vectors and multiplied by scalars.

Vector Addition

The operation that combines two vectors to produce another vector.

Vector Space

A collection of vectors with vector addition and scalar multiplication that obey the required algebraic rules.

Zero Subspace

The subspace containing only the zero vector.

Zero Vector

The vector that leaves every vector unchanged when added to it.

Zero Vector Space

A vector space containing only the zero vector

Column Vector

A matrix with one column.

Diagonal Entries of a Matrix

The entries running from the upper-left corner to the lower-right corner, where the row position and column position are the same.

Diagonal Matrix

A square matrix whose entries outside the main diagonal are all zero.

Matrix

A rectangular arrangement of entries organized into rows and columns.

Matrix Addition

Adding two matrices of the same size by adding entries in matching positions.

Row Vector

A matrix with one row.

Square Matrix

A matrix with the same number of rows and columns

Symmetric Matrix

A square matrix that remains unchanged when its rows and columns are exchanged.

Trace

The sum of the diagonal entries of a square matrix.

Transpose

The matrix obtained by exchanging the rows and columns of another matrix.

Upper Triangular Matrix

A square matrix whose entries below the main diagonal are all zero.

Zero Matrix

A matrix in which every entry is zero.

Degree of a Polynomial

The greatest exponent of the variable that has a nonzero coefficient.

Lagrange Interpolation Formula

A method for constructing the unique polynomial of sufficiently small degree that passes through a specified collection of points with different input values.

Lagrange Polynomials

Special polynomials used in Lagrange interpolation, each equal to one at one chosen input value and zero at the other chosen input values.

Polynomial

An expression made from constants and nonnegative whole-number powers of a variable, combined through addition and multiplication.

Standard Basis for the Space of n-Tuples over a Field (Fⁿ)

The set of basic column vectors in which exactly one entry is one and every other entry is zero.

Standard Basis for the Space of Polynomials of Degree at Most n over a Field Pn(F)

The set consisting of the constant polynomial, the variable, its square, and continuing through the polynomial whose degree is n.

Zero Polynomial

The polynomial whose coefficients are all zero.

Chain/Nest/Tower

A subset of a partially ordered set in which every pair of elements can be compared.

Hausdorff Maximal Principle

The principle stating that every partially ordered set contains a maximal chain

Maximal Element of a Family of Sets

A set in the family that is not properly contained in any other set in that family.

n-Tuple

An ordered list of n elements.

Sequence

An ordered list of elements, usually indexed by the positive whole numbers.

dimension

The number of vectors in any basis of a finite-dimensional vector space. It describes how many independent directions are needed to represent every vector in the space.

Parallel Vectors

Vectors that lie along parallel lines; they have the same or opposite direction.

Parallelogram Law of Vector Addition

When two vectors begin at the same point and form adjacent sides of a parallelogram, their sum is represented by the diagonal that begins at their shared starting point