GATE Linear Algebra formula sheet

Applies to any subset W of a vector space V. If the zero vector is missing, W fails immediately.

W1 and W2 are finite-dimensional subspaces of a vector space V. The intersection and the sum W1+W2 (all u+v with u in W1, v in W2) are always subspaces. The union is a subspace exactly when one of the two contains the other.

Holds for every square real matrix, and the split is unique.

Square real $Q$. Columns (and rows) form an orthonormal set.

Applies to any $m\times n$ matrix $A$ after row reduction; the rank is the same in every echelon form.

Equivalent statement: $\mathbf{b}$ lies in the column space of $A$. Always $\mathrm{rank}(A)\le \mathrm{rank}([A\mid \mathbf{b}])\le \mathrm{rank}(A)+1$.

Always valid for a 2 by 2 matrix.

A is n by n and i is any fixed row. M_ij is the determinant of the (n-1) by (n-1) matrix left after deleting row i and column j. The same works along any column. Expand along the row or column with the most zeros.

Line through the origin along $\mathbf{a}\ne 0$; $\mathbf{b}$ and $\mathbf{a}$ are in ${\mathbb{R}}^n$, $P$ is $n\times n$ of rank $1$, and $\mathbf{p}=P\mathbf{b}$.

$A$ is $m\times n$ with linearly independent columns (so $A^{T}A$ is invertible); $P$ is $m\times m$ and projects onto the column space of $A$.