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The sum of two vectors results in a third vector within the same vector space.

A vector can be multiplied by a scalar (a real number), resulting in a new vector that has been scaled by that number.

A subset of a vector space that itself is a vector space under the addition and scalar multiplication defined on the larger space.

A set of vectors is linearly dependent if at least one vector in the set can be written as a linear combination of the others.

A set of vectors is linearly independent if the only linear combination that equals the zero vector is the trivial combination (all scalars are zero).

A set of vectors in a vector space V that are linearly independent and span V.

The set of all possible linear combinations of a given set of vectors.

Defined as the number of vectors in a basis for the vector space, indicating the degrees of freedom or the minimum number of coordinates needed to specify each point within the space.

A unique vector in a vector space that, when added to any vector in the space, returns the original vector (acts as an additive identity).

A sum of scalar multiples of vectors. Formally, if $\{\vec{v}_1, \vec{v}_2, ..., \vec{v}_k\}$ are vectors and $\{a_1, a_2, ..., a_k\}$ are scalars, then $a_1\vec{v}_1 + a_2\vec{v}_2 + ... + a_k\vec{v}_k$ is a linear combination.

A function between two vector spaces that preserves the operations of vector addition and scalar multiplication.

The set of all vectors that map to the zero vector under a linear transformation T, denoted as $\text{Ker}(T)$.

The set of all possible outputs of a linear transformation T, denoted as $\text{Im}(T)$ or $\text{Range}(T)$.

A bijective (one-to-one and onto) linear transformation between two vector spaces. An isomorphism indicates the two vector spaces are structurally the same, i.e., they are isomorphic.

Given a vector space with a specific basis, the coordinate vector is an ordered list of scalars that represents a vector in terms of the given basis.

For a linear transformation T, a scalar $\lambda$ is an eigenvalue if there is a non-zero vector $\vec{v}$, such that $T(\vec{v}) = \lambda\vec{v}$.

Given a linear transformation T and an eigenvalue $\lambda$, an eigenvector is a non-zero vector $\vec{v}$ that satisfies the equation $T(\vec{v}) = \lambda\vec{v}$.

The set of all linear functionals on a vector space V, often denoted as $V^*$ or $V'$, where a linear functional is a linear transformation from V to its field of scalars.

Two vectors are orthogonal if their dot product (inner product) is zero.

An algorithm for orthogonalizing a set of vectors in an inner product space, which also can be used to produce an orthonormal set of vectors.

A matrix A that when multiplied by the coordinate vector of $\vec{v}$ in the domain space relative to a basis $B$, yields the coordinate vector of the transformed vector $T(\vec{v})$ in the codomain space relative to a basis $C$.

Given a subspace W of V, the orthogonal complement, denoted $W^\perp$, is the set of all vectors in V that are orthogonal to every vector in W.

A construction that takes two vector spaces V and W, and produces a new vector space V ⊗ W, whose elements are formal linear combinations of tensors, each of which is the 'product' of a vector from V and a vector from W.