Linear Algebra Practice Exam Vocabulary

A set of vocabulary flashcards based on linear algebra lecture notes covering vector spaces, linear independence, rank, nullspace, and basis properties.

Linear Combination

An expression for a vector $b$ constructed by summing scalar multiples of other vectors, such as $b = c_1u_1 + c_2u_2 + ... + c_nu_n$.

Reduced Row Echelon Form (RREF)

A form of a matrix used to determine the existence of solutions, where leading $1$'s identify pivots and zeros appear in columns below and above each leading $1$.

Rank(A)

The dimension of the row space or column space of matrix $A$, equivalent to the number of leading $1$'s in its reduced row echelon form.

Nullspace (N(A))

The solution space to the homogeneous equation $Ax = 0$, where its dimension is equal to the number of free variables in the system.

Linearly Independent

A property of a set of vectors where the equation $c_1P_1 + c_2P_2 + c_3P_3 = 0$ has only the trivial solution $c_1 = c_2 = c_3 = 0$.

Basis

A set of vectors $S$ for a vector space $V$ that satisfies two conditions: the vectors are linearly independent and they span $V$.

Spanning Set

A set of vectors $S$ such that every vector in the vector space can be written as a linear combination of vectors in $S$.

Column Space (C(A))

The subspace spanned by the column vectors of a matrix $A$, where the pivot columns of the original matrix form a basis.

Symmetric Matrix Space ($S_{3,3}$)

The vector space of all $3 \times 3$ symmetric matrices, which has a dimension of $6$.

Polynomial Space (P)

The space of all polynomial functions, which is considered an infinite-dimensional vector space.

Coordinate Vector ($[V]_B$)

The vector of scalars $c_1, c_2, ..., c_n$ used to express vector $v$ as a linear combination of the vectors in a specific basis $B$.

Vector Space Axioms

The rules that a set must satisfy to be considered a vector space, such as closure under addition and scalar multiplication; sets like "all fifth-degree polynomials" or ordered pairs where $y \ge 0$ often fail these.

Union of Subspaces

The set containing all elements that belong to either subspace $W$ or $U$; this set is generally not a subspace of the parent vector space $V$.

Linearly Dependent

A set of vectors where at least one vector can be written as a linear combination of the others, or where a nontrivial solution to $c_1v_1 + ... + c_nv_n = 0$ exists.

Determinant Test for Independence

A shortcut where if the determinant of a square matrix formed by vectors is non-zero, the columns are linearly independent; if it is $0$, they are linearly dependent.

Row Operations on Column Dependency

Elementary row operations performed on a matrix $A$ that do not change the linear dependency relationships among its columns.

Row Space Equality

A property where two row-equivalent matrices $A$ and $B$ share the same row space ($R(A) = R(B)$), even though their column spaces may differ.