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Linear Algebra

Part 1 Lecture 1

This is the first in a series of articles providing an introduction to linear algebra.

Scalar

A scalar is a single number.

This is an example of a scalar: 5

This is also a scalar: 7.5

Here is another scalar: 39

Vector

A vector is a row or column of numbers.

Here is an example of a row vector:

The following is a column vector:

Vectors are thought of as n-tuples: (a₁, a₂, … a_n)

This is a 3-tuple: (x, y, z)

The order of the components is important. For example: (x, y, z) is not equal to (x, z, y)

The components can be called cells, entries or dimensions.

An example: For the (x, y, z) vector, x is the first dimension, y is the second dimension, and z is the third dimension. Or you could say x is the first entry, y is the second entry, etc.

Scalar Multiplication

Scalar Multiplication means multiplying a vector by a scalar, producing a scaled vector:

5 ( 1, 2, 3 ) = ( 5, 10, 15 )

This is thought of as scaling a vector.

Scalar Product

(Inner Product, Dot Product)

The Scalar Product, on the other hand, is a multiplication of vectors that always produces a scalar. The product of this process is always a single number.

This is also called the Inner Product, or Dot Product.

The mathematical symbol for this kind of multiplication is the “middle dot” (·) symbol.

It is important to know that scalar multiplication and the scalar product are not the same. Scalar multiplication always has a scalar multiplier, while the scalar product always has vector multipliers (no scalar multipliers).

With the scalar product, only the product is a scalar – the multipliers are always vectors.

The Scalar Product is produced by processing two vectors that have the same number of components (cells). The two vectors are “multiplied” together to produce a single number (a scalar).

This “multiplication” process is actually quite interesting. It consists of multiplying together the corresponding scalar components (cells) of each vector, and then adding those products together to yield a single number which we call the “scalar product.”

For example, the scalar product of ( 2, 3 ) and ( 5, 6 ) is:

( 2, 3 ) · ( 5, 6 ) = 2 × 5 + 3 × 6 = 28

Notice that the symbol for this process is a dot (·), not the multiplication sign (×).

We will refer to this process as the inner product or dot product in these pages. In advanced calculus, the term scalar product is used for this process. In linear algebra, the terms scalar product, inner product and dot product mean the same thing, and are used interchangeably.

The term inner product is descriptive of this process, since the products of the internal parts (components) of the vectors are summed together.

There is something else called an outer product which does not sum the products of components together, producing a vector instead of a scalar.

Our primary interest in this discussion will be with the scalar product (i.e., inner product, dot product), which we will see is used inside matrices to perform matrix multiplication.

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