# Vector Operations

v[[i]] or Part[v,i] | give the i element in the vector v |

c v | scalar multiplication of c times the vector v |

u.v | dot product of two vectors |

Norm[v] | give the norm of v |

Normalize[v] | give a unit vector in the direction of v |

Standardize[v] | shift v to have zero mean and unit sample variance |

Standardize[v,f_{1}] | shift v by and scale to have unit sample variance |

Basic vector operations.

This is a vector in three dimensions.

Out[1]= | |

This gives a vector

in the direction opposite to

with twice the magnitude.

Out[2]= | |

This reassigns the first component of

to be its negative.

Out[3]= | |

This gives the dot product of

and

.

Out[4]= | |

This is the norm of

.

Out[5]= | |

This is the unit vector in the same direction as

.

Out[6]= | |

This verifies that the norm is 1.

Out[7]= | |

Transform

to have zero mean and unit sample variance.

Out[8]= | |

This shows the transformed values have mean 0 and variance 1.

Out[9]= | |

Two vectors are orthogonal if their dot product is zero. A set of vectors is orthonormal if they are all unit vectors and are pairwise orthogonal.

Projection[u,v] | give the orthogonal projection of u onto v |

Orthogonalize[{v_{1},v_{2,}...}] | generate an orthonormal set from the given list of vectors |

Orthogonal vector operations.

This gives the projection of

onto

.

Out[10]= | |

is a scalar multiple of

.

Out[11]= | |

is orthogonal to

.

Out[12]= | |

Starting from the set of vectors

, this finds an orthonormal set of two vectors.

Out[13]= | |

When one of the vectors is linearly dependent on the vectors preceding it, the corresponding position in the result will be a zero vector.

Out[14]= | |