A vector space $V$ must have closure under addition and scalar multiplication, plus:
Happy studying, and may your vectors always be linearly independent. lecture notes for linear algebra
An inner product on vector space (V) satisfies: A vector space $V$ must have closure under
| Concept | Formula / Condition | |---------|----------------------| | Linear system solution | Gaussian elimination | | Dot product | (\mathbfu\cdot\mathbfv = \sum u_i v_i) | | Matrix multiplication | ((AB) ij = \sum_k a ikb_kj) | | Linear independence | (c_1\mathbfv_1+\dots=0) ⇒ all (c_i=0) | | Determinant (2×2) | (ad-bc) | | Eigenvalue equation | (\det(A-\lambda I)=0) | | Diagonalization | (A = PDP^-1) | | Orthonormal basis | (\mathbfq_i\cdot\mathbfq j = \delta ij) | | Least squares | (A^T A \mathbfx = A^T \mathbfb) | plus: Happy studying
No article on is complete without the two titans: the inverse and the determinant.
Given basis (\mathbfv_1,\dots,\mathbfv_n), produce orthonormal basis (\mathbfu_1,\dots,\mathbfu_n): [ \mathbfu_1 = \frac\mathbfv_1\mathbfv_1, \quad \mathbfw_2 = \mathbfv_2 - (\mathbfv_2\cdot\mathbfu_1)\mathbfu_1, \quad \mathbfu_2 = \frac\mathbfw_2 ] Repeat.