Vector Spaces and Subspaces: every key term you need (+ practice quiz)
25 flashcard terms for Linear Algebra Topic 3, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
A set with addition and scalar multiplication satisfying ten axioms: closure, commutativity, associativity, a zero vector, additive inverses, and four compatibility rules for scalars.
Field of scalars
The number system supplying the scalars of a vector space, most often the real numbers or the complex numbers; changing the field changes which spaces and dimensions arise.
Zero vector axiom
The requirement that a vector space contain an element leaving every vector unchanged under addition. It is unique, and multiplying any vector by the scalar zero produces it.
Subspace
A nonempty subset of a vector space that contains the zero vector and is closed under both addition and scalar multiplication; it is then a vector space in its own right.
Subspace test
A subset qualifies as a subspace exactly when it contains zero and every linear combination of two of its members stays inside the subset.
Trivial subspace
The subspace consisting of the zero vector alone. Together with the whole space it forms the pair of subspaces that every vector space possesses.
Span
The set of all linear combinations of a given list of vectors. It is always a subspace, indeed the smallest subspace containing the listed vectors.
Spanning set
A list of vectors whose span is the entire space. Adding vectors to a spanning set keeps it spanning, while removing vectors may destroy the property.
Linear combination
A sum of scalar multiples of finitely many vectors. Coefficients may be any scalars, including zero, and only finitely many terms are allowed.
Column space
The span of the columns of a matrix, a subspace of the target space. A system A x = b is consistent exactly when b lies in this subspace.
Row space
The span of the rows of a matrix. Row operations leave it unchanged, so the nonzero rows of any echelon form of the matrix supply a basis for it.
Null space
The set of all vectors sent to zero by a matrix, that is the solution set of the homogeneous system. It is a subspace of the domain space.
Function space
A vector space whose elements are functions on a fixed set, with addition and scaling defined pointwise; continuous functions on an interval form a familiar example.
Polynomial space
The space of polynomials of degree at most n together with the zero polynomial. It is closed under addition and scaling and has dimension n plus one.
Matrix space
The set of all matrices of a fixed size, made into a vector space by entrywise addition and scaling; the symmetric matrices form a subspace of the square case.
Sequence space
A vector space of infinite sequences of scalars with termwise operations. It illustrates that vector spaces need not be finitely generated.
Sum of subspaces
The set of all vectors formed by adding one element of each of two subspaces. It is itself a subspace and is the smallest one containing both.
Direct sum
A sum of two subspaces whose intersection is only the zero vector, so every element of the sum decomposes into summands in exactly one way.
Intersection of subspaces
The set of vectors lying in both of two subspaces. It is always a subspace, unlike the union, which usually fails closure under addition.
Closure under addition
The property that adding any two members of a set produces another member. Failure of this property is the usual reason a plausible subset is not a subspace.
Affine subset
A translate of a subspace by a fixed vector. Unless the translation is by zero it misses the origin and so is not itself a subspace.
Coordinate space
The space of ordered tuples of n scalars with componentwise operations. Every finite dimensional space over the same field looks structurally identical to one of these.
Quotient construction
The space whose elements are the translates of a fixed subspace, added and scaled through representatives; it measures what is left once the subspace is collapsed to zero.
Generating a subspace from a set
Given any subset of a vector space, the collection of its finite linear combinations is a subspace, the smallest one containing the original subset.
Cancellation in a vector space
A consequence of the axioms stating that if u plus w equals v plus w then u equals v; it follows from adding the additive inverse of w to both sides.