Key concepts#

Vectors#

  • Vectors are ordered lists of numbers that can also be viewed geometrically (as arrows)

Vector operations#

  • Length of a vector: \(||\bar{x}|| = \sqrt{\sum_{i=1}^N x_i^2} = \sqrt{\bar{x}\cdot \bar{x}}\)

  • Unit vectors have length = 1. To make a unit vector version of \(\bar{x}\): \(\hat{x} = \frac{\bar{x}}{||\bar{x}||}\)

  • Zero vectors (\(\bar{0}\)) have all components equal to 0

  • Scalar multiplication of a vector: \( a\bar{x} = \begin{bmatrix} ax_1 \\ ax_2 \\ \vdots \\ ax_N \end{bmatrix} \)

  • Vector addition: \(\bar{x} + \bar{y} = \begin{bmatrix} x_{1} + y_1 \\ x_{2} + y_2\\ \vdots \\ x_{N} + y_N \end{bmatrix}\)

  • Dot product: \(\bar{x} \cdot \bar{y} = \sum_{i=1}^N x_iy_i\)

Vector spaces#

  • Linear combination: building one vector as the sum of scaled multiples of other vectors

  • The span of a set of vectors, \(Span\{\bar{v}_1, \bar{v}_2, ...\}\), is the set of all linear combinations of those vectors

  • A set of non-zero vectors is linearly dependent if one member of the set is a linear combination of the others, otherwise they are linearly independent.

  • A basis for a given space is a set of linearly independent vectors that spans the space.

  • A vector space V is a collection of vectors for which 2 operations are defined, addition and multiplication by a scalar, that satisfy the below axioms for all vectors in V. If \(\bar{x}, \bar{y},\) and \(\bar{z}\) are any vectors in V:

    1. \(\bar{x} + \bar{y}\) is in V

    2. \(\bar{x} + \bar{y} = \bar{y} + \bar{x}\)

    3. \((\bar{x} + \bar{y}) + \bar{z} = \bar{x} + (\bar{y} + \bar{z})\)

    4. Zero vector exists in V such that \(\bar{x} + \bar{0} = \bar{x}\)

    5. The inverse \(-\bar{x}\) exists such that \(\bar{x} + (-\bar{x}) = \bar{0}\)

    6. \(c\bar{x}\) is in V

    7. \(c(\bar{x} + \bar{y}) = c\bar{x} + c\bar{y}\)

    8. \((c+d)\bar{x} = c\bar{x} + d\bar{x}\)

    9. \(c(d\bar{x}) = (cd)\bar{x})\)

    10. \(1\bar{x} = \bar{x}\)