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13 results for “vector”

Formulae (13)

\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)

Vector of partial derivatives pointing in the steepest ascent.

\sigma(z_i) = \frac{e^{z_i}}{\sum_{j=1}^{K} e^{z_j}}

Converts a vector of scores into a probability distribution.

\cos\theta = \frac{A \cdot B}{\|A\|\|B\|}

Measures similarity between two vectors by the angle between them.

\vec{a}\cdot\vec{b} = \sum_i a_i b_i = |a||b|\cos\theta

Scalar product of two vectors.

|\vec{a}\times\vec{b}| = |a||b|\sin\theta

Magnitude of the vector perpendicular to two vectors.

|\vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}

Length of a vector.

A\vec{v} = \lambda\vec{v}

Defines eigenvalues and eigenvectors of a matrix.

\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2}\vec{b}

Projection of vector a onto vector b.

\mathbf{a}\cdot\mathbf{b} = \sum a_i b_i

Sum of componentwise products of two vectors.

\|\mathbf{v}\| = \sqrt{\sum v_i^2}

Length of a vector.

\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{\|a\|\|b\|}

Angle from the dot product and magnitudes.

\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})

Volume of the parallelepiped spanned by three vectors.

\hat{\mathbf{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|}

Vector of length one in the same direction.