Linear Algebra Solver

For Linear Independence: Check if the rank of the matrix formed by the vectors equals the number of vectors.

\(\text{rank}(A) = n\)
(independent if true)

Where:
– \(A\) = matrix with vectors as columns
– \(n\) = number of vectors
– \(\text{rank}(A)\) = number of pivots in reduced row echelon form (RREF)

For Basis Finder: Extract non-zero rows from RREF for row space or span; use pivot columns for column space; solve for free variables for null space.
For column space example:

\(B = \{ \mathbf{v}_i \mid i \in \text{pivot columns} \}\)

Where:
– \(B\) = set of basis vectors
– \(\mathbf{v}_i\) = original input vectors corresponding to pivot positions

For Span: The span is all linear combinations of the vectors; a basis is obtained from the non-zero rows of the RREF.

\(\text{span}\{\mathbf{v}_1, \dots, \mathbf{v}_n\} = \left\{ \sum_{i=1}^n c_i \mathbf{v}_i \;\middle|\; c_i \in \mathbb{R} \right\}\)

Where:
– \(\mathbf{v}_i\) = input vectors
– \(c_i\) = arbitrary scalar coefficients

For Dimension:
Dimension equals rank for span, row space, or column space; equals number of columns minus rank for null space.

\(\dim(V) = \text{rank}(A)\)
(for span / row space / column space)

Where:
– \(V\) = vector space under consideration
– \(A\) = associated matrix (rows or columns depending on space type)

For Subspace Checker: A set \(S\) is a subspace if it satisfies the subspace axioms:

\(0 \in S, \quad \mathbf{u} + \mathbf{v} \in S, \quad c\mathbf{u} \in S\)
(for all \(\mathbf{u}, \mathbf{v} \in S\) and all scalars \(c\))

Where:
– \(S\) = the given set
– \(\mathbf{u}, \mathbf{v}\) = any two elements in the set
– \(c\) = any scalar

For Gram-Schmidt Orthogonalization:
The classical process (orthogonal version):

\(\mathbf{u}_k = \mathbf{v}_k – \sum_{j=1}^{k-1} \frac{\langle \mathbf{v}_k, \mathbf{u}_j \rangle}{\langle \mathbf{u}_j, \mathbf{u}_j \rangle} \mathbf{u}_j\)

For orthonormal basis, normalize each vector:
\(\mathbf{e}_k = \frac{\mathbf{u}_k}{\|\mathbf{u}_k\|}\)

Where:
– \(\mathbf{v}_k\) = k-th input vector
– \(\langle \cdot, \cdot \rangle\) = dot (inner) product
– \(\|\cdot\|\) = Euclidean norm

For Orthogonality Checker: A set is orthogonal if off-diagonal entries of the Gram matrix are zero. For orthonormal, diagonal entries must be 1.

\(G_{ij} = \langle \mathbf{v}_i, \mathbf{v}_j \rangle = 0 \quad (i \neq j)\)

Where:
– \(G\) = Gram matrix
– \(\mathbf{v}_i, \mathbf{v}_j\) = vectors in the set

For Vector Projection: Projection of \(\mathbf{u}\) onto a single vector \(\mathbf{v}\):

\(\mathrm{proj}_{\mathbf{v}} \mathbf{u} = \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{\langle \mathbf{v}, \mathbf{v} \rangle} \mathbf{v}\)

For projection onto a subspace with orthogonal basis \(\{\mathbf{b}_1, \dots, \mathbf{b}_m\}\):

\(\mathrm{proj} \mathbf{u} = \sum_{j=1}^m \frac{\langle \mathbf{u}, \mathbf{b}_j \rangle}{\langle \mathbf{b}_j, \mathbf{b}_j \rangle} \mathbf{b}_j\)

Where:
– \(\mathbf{u}\) = vector being projected
– \(\mathbf{v}\) or \(\mathbf{b}_j\) = target vector or basis vectors

For Change of Basis: New coordinates in basis \(B’\) from old coordinates in basis \(B\):

\([\mathbf{x}]_{B’} = P^{-1} [\mathbf{x}]_B\)

Where:
– \(B, B’\) = old and new bases
– \(P\) = change-of-basis (transition) matrix whose columns are the old basis vectors expressed in the new basis

For Rank-Nullity Theorem: Fundamental relationship for any matrix:

\(\text{rank}(A) + \dim(\ker A) = n\)

Where:
– \(A\) = \(m \times n\) matrix
– \(n\) = number of columns
– \(\ker A\) = null space of \(A\)
– \(\dim(\ker A)\) = nullity of \(A\)

These formulas cover the core computations performed by the Linear Algebra Solver tool across all supported calculator types. They are implemented using high-precision Gaussian elimination (RREF), dot products, and normalization where applicable, with numerical tolerance control for floating-point reliability.

Matrix Computations — Golub & Van Loan

Linear Algebra — Hoffman & Kunze