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- The (orthogonal) projection of a vector onto a subspace is the point in the subspace closest to the vector.#23943
- The permutation expansion for determinants#23727
- The pivot columns of a matrix are a basis for the column space.#23877
- The preimage of a vector is a translation of the kernel of the linear transformation#24031
- The product of square matrices is nonsingular if and only if each factor is nonsingular.#23524
- The product of upper/lower triangular matrices is upper/lower triangular.#23564
- The projection of a vector which is in a subspace is the vector itself.#23942
- The QR decomposition of a nonsingular matrix exists.#23632
- The range/image of a linear transformation is a subspace.#24056
- The range of a linear transformation is a subspace#24036
- The range spaces of powers of a linear transformation form a descending chain#24063
- The rank of a matrix equals number of pivots in a reduced row echelon form.#23676
- The rank of a matrix equals the rank of the linear transformation it represents.#23677
- The rank plus the nullity of a linear transformation equals the dimension of the domain.#24064
- The reduced row-echelon form of a matrix determines which subset of a spanning set is a basis.#23986
- The row space and the column space of a matrix have the same dimension.#23678
- The row space of a matrix is a vector space#24108
- The set containing only 0 is a vector space.#24096
- The set of all functions on a set is a vector space.#24100
- The set of all polynomials is a vector space.#24103
- The set of all polynomials of degree at most n is a vector space.#24104
- The set of all sequences is a vector space.#24106
- The set of linear transformations between two vector spaces is a vector space.#24112
- The set of m by n matrices is a vector space.#24105
- The solutions of a homogeneous system are the pre-image (of 0) of a linear transformation.#23868
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