Linear independence axiomatic properties:

• singleton set containing one non-zero vector $\{v\}$ is always linearly independent because $c v = 0 \implies c = 0$.
• The empty set is independent by definition, not dependent.
• In linear algebra, the empty set $\emptyset$ is defined as linearly independent vacuously, because it contains no vectors that can form a non-trivial linear combination equaling the zero vector.
• Sets containing the zero vector or collinear vectors are linearly dependent.

Fundamental properties of linear transformations:

• The null space (kernel) is a subspace of domain $V$, not codomain $W$.
• The range (rank) is a subspace of codomain $W$, not domain $V$.
• By linearity, $T(0_V) = T(0 \cdot 0_V) = 0 \cdot T(0_V) = 0_W$. Every linear transformation maps the zero vector of domain space $V$ to the zero vector of codomain space $W$.
• Linear transformations map the zero vector to the zero vector, not the identity matrix.

Step-by-step matrix inversion:

• Erroneous adjoint calculation.
• Determinant $\det(A) = 0(4 - 6) - 2(2 - 6) + 4(6 - 12) = 0 + 8 - 24 = -16$. The cofactor matrix is: $C = \begin{pmatrix} -2 & 4 & -6 \\ 10 & -12 & 6 \\ -12 & 8 & -4 \end{pmatrix}$. Transposing to get adjoint: $\operatorname{adj}(A) = \begin{pmatrix} -2 & 10 & -12 \\ 4 & -12 & 8 \\ -6 & 6 & -4 \end{pmatrix}$. Dividing by $-16$ gives $A^{-1} = \begin{pmatrix} 1/8 & -5/8 & 3/4 \\ -1/4 & 3/4 & -1/2 \\ 3/8 & -3/8 & 1/4 \end{pmatrix}$.
• Has $3/8$ instead of $3/4$ at position (2,2).
• Distractor with zeroes.

Axiomatic rank properties of matrices:

• An invertible matrix has full rank equal to its order $n$, never $0$.
• The $n \times n$ identity matrix has rank $n$.
• non-singular square matrix has rank $n \gt 0$.
• The rank of a matrix is defined as the maximum number of linearly independent row/column vectors. The rank of a matrix is strictly greater than or equal to $1$ for any non-zero matrix. The only matrix whose rank is identically $0$ is the zero matrix (null matrix).