Scalars and Vectors

In the last lesson we looked at matrices when you may have noticed that major cities are two dimensional

objects by definition as they have rows and columns.

However there are specific cases of matrices which are also widely used one of them is overwhelmingly

common a matrix with one row and one column.

Naturally it contains a single element that is called a scalar.

In fact all numbers we know from algebra are referred to as scalars in linear algebra.

For instance the number 15 is a scalar.

So are 1 to minus 5 PI and so on.

Since scalars are objects with no dimensions they have important properties.

We are going to examine later on.

For now we just need to introduce the basic terminology in order to be able to dive into linear algebraic

operations.

All right.

Next on our list are vectors.

Vectors are also very common objects in linear algebra.

They said somewhere between scalars and matrices as they have one dimension here is a vector with three

elements five minus 2 and 4.

Notice that this vector is nothing more than a three by one matrix its single dimension is the number

of rows it has since a scalar is basically a number.

A vector is practically the simplest linear algebraic object due to the many applications of vectors.

They have this special name and are often examined as a separate object.

Moreover it is more common to view a matrix as a collection of vectors rather than a vector as a special

case of a matrix.

OK generally there are two types of vectors row vectors and column vectors.

We just explored a column vector.

Logically we can have row vectors like this one.

3 4 5 8 often we are interested in the number of elements of vector contains We refer to that number

as the length of the vector.

The first vector we saw had a length of three while the second one a length of four.

OK great.

In the general case of vectors length can be designated as m.

So a column vector would have dimensions m by one and the row vector one by M regarding the notation.

The elements of a vector x would be x 1 x 2 until x M.

Let’s summarize.

We have major cities that have two dimensions.

M by n vectors that have a single dimension and by one and scalars that have no dimensions and are one

by one in the consequent lectures we will learn how to add subtract and multiply matrices vectors and

scalars which will make these concepts even more intuitive.

Thanks for watching.