math111_logo Vector and Matrix

1. Euclidean vector and euclidean space

The n-dimensional (real) euclidean space Rn is the collection of n-tuples of numbers, called vectors. Vectors can be either presented horizontally,

x = (x1, x2, ..., xn)

or vertically.

x = [ x1 ]
x2
:
xn

The only thing you need to keep in mind is that, when vectors are mixed with matrices, it is often more convenient to use vertical vectors.

The number n is the dimension of the vector x. The numbers x1, x2, ..., xn are called the coordinates of x. Strictly speaking, these are the coordinates with respect to the standard basis vectors

e1 = (1, 0, ..., 0),

e2 = (0, 1, ..., 0),

:

en = (0, 0, ..., 1).

We also mention the special zero vector

0 = (0, 0, ..., 0).

Example The 1-dimensional euclidean space

R1 = {(x): x is a real number}

can be identified with all real numbers R and be visualized as a straight line.

The 2-dimensional euclidean space

R2 = {(x1, x2): x1, x2 are real numbers}

can be visualized as the coordinate plane.

picture of R^2

The 3-dimensional euclidean space

R3 = {(x1, x2, x3): x1, x2, x3 are real numbers}

can be visualized as the world we are living in.

picture of R^3

The 0-dimensional euclidean space R0 consists of a single point - the origin.

The example illustrates that the euclidean vectors can be geometrically presented (in a coordinate system) as an arrow starting from the origin. If an arrow does not start from the origin, we need to parallel shift it so that it begins at the origin.

vectors start at the origin

Vectors of the same dimension can be added

(x1, x2, ..., xn) + (y1, y2, ..., yn) = (x1 + y1, x2 + y2, ..., xn + yn).

A number (scalar) can also be multiplied to vectors

c (x1, x2, ..., xn) = (cx1, cx2, ..., cxn).

The addition and scalar multiplication can also be combined to form linear combinations c1x1 + c2x2 + ... + ckxk.

Example Let u = (1, 2), v = (3, 4), w = (5, 6), x = (1, 0, -3), y = (2, -1, 5). Then

u + v = (1 + 3, 2 + 4) = (4, 6)
3u = (3×1, 3×2) = (3, 6)
u + v + w = (1 + 3 + 5, 2 + 4 + 6) = (9, 12)
3u - 4v + w = (3×1 - 4×3 + 5, 3×2 - 4×4 + 6) = (-4, -4)

x + y = (1 + 2, 0 - 1, -3 + 5) = (3, -1, 2)
-2x = (-2×1, -2×0, -2×(-3)) = (-2, 0, 6)
-2x + 3y = (-2×1 + 3×2, -2×0 + 3×(-1), -2×(-3) + 3×5) = (4, -3, 21)

Moreover, because the dimension of u, v, w is different from the dimension of x, y, expressions such as u + x, 2v + 3y are meaningless.

Geometrically, the addition is described by parallelogram, and the scalar multiplication is the stretching/shrinking and sometimes reversing (for negative scalars) of vectors.

vectors operations

This picture shows the linear combinations of two (nonparallel) vectors. It suggests that all the linear combinations of the two vectors form a plane passing through the origin.

linear combination

Finally, the operations have the following obvious properties.

commutative
u + v = v + u
associative
(u + v) + w = u + (v + w)
zero
u + 0 = u = 0 + u
distributive
r(u + v) = ru + rv
distributive
(r + s)u = ru + su
associative
(rs)u = r(su)

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[Extra Vector addition is given by parallelogram]