Linear Independence, Basis and Dimension: every key term you need (+ practice quiz)
25 flashcard terms for Linear Algebra Topic 4, 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 list of vectors is independent when the only linear combination equal to zero is the one with every coefficient zero; no vector then lies in the span of the others.
Linear dependence
A list is dependent when some nontrivial combination of its members equals zero, which happens exactly when one vector is a combination of the earlier ones.
Dependence relation
An explicit equation with not all coefficients zero showing that a list of vectors sums to zero; its coefficients are read off from the null space of the matrix of those vectors.
Independence test by row reduction
Place the vectors as columns and reduce. The list is independent exactly when every column is a pivot column, that is when the homogeneous system has only the trivial solution.
Too many vectors are dependent
Any list of more than n vectors inside an n dimensional space must be dependent, since the associated homogeneous system has more unknowns than equations.
Sets containing the zero vector
Any list that includes the zero vector is automatically dependent, because a nonzero coefficient on that vector alone produces a nontrivial relation.
Basis
An independent list that spans the space. It is simultaneously as large as an independent list can be and as small as a spanning list can be.
Standard basis
The basis of coordinate space consisting of the columns of the identity matrix, each having a single one and zeros elsewhere.
Spanning set theorem
A spanning list can be pruned to a basis by discarding vectors that lie in the span of the others; the remaining vectors still span the same subspace.
Extension to a basis
Any independent list in a finite dimensional space can be enlarged to a basis by adjoining vectors from any spanning set until the whole space is reached.
Unique representation property
Relative to a fixed basis every vector has exactly one expression as a linear combination of the basis vectors; existence comes from spanning and uniqueness from independence.
Coordinate vector
The list of coefficients expressing a vector in a given ordered basis. The map sending a vector to its coordinates is a one to one and onto linear correspondence.
Change of basis matrix
The matrix whose columns are the coordinate vectors of one basis with respect to another; it converts coordinates from the first basis to the second and is always invertible.
Dimension
The number of vectors in any basis of a space. The invariance theorem guarantees this count does not depend on which basis is chosen.
Invariance of dimension
The theorem that any two bases of the same finite dimensional space have equal length, proved by showing a longer independent list inside a shorter span is impossible.
Finite dimensional space
A space spanned by some finite list of vectors. Every such space has a basis and a well defined dimension, and every subspace of it is again finite dimensional.
Infinite dimensional space
A space with no finite spanning list, such as the space of all polynomials or the space of continuous functions on an interval.
Subspace dimension bound
A subspace of a finite dimensional space has dimension no larger than the whole space, with equality only when the subspace is the entire space.
Basis theorem for a space of known dimension
In a space of dimension n, any independent list of n vectors is automatically a basis, and so is any spanning list of n vectors.
Basis of a column space
The pivot columns of the original matrix, not the reduced one, form a basis for the column space, since row reduction preserves dependence relations among columns.
Basis of a null space
The vectors obtained from parametric vector form, one for each free variable, are independent and span the null space, so they form a basis for it.
Maximal independent list
An independent list that cannot be enlarged inside the space without losing independence; in a finite dimensional space such a list is exactly a basis.
Minimal spanning list
A spanning list from which no vector can be deleted while still spanning; in a finite dimensional space these lists are precisely the bases.
Ordered basis
A basis with a fixed order imposed on its vectors, which is required before coordinate vectors and matrix representations can be written down unambiguously.
Dimension of a sum of subspaces
The dimension of a sum equals the sum of the dimensions minus the dimension of the intersection, so the sum is direct exactly when the dimensions simply add.