Understanding Scalars Vectors and Matrices in Linear Algebra

Scalars are single numbers and are an example of a 0th -order tensor. Mathematically, it is mandatory to explain the set of values to which a scalar belongs. There are different sets of numbers of interests within the concept of machine learning. 

The letter N represents the set of positive integers. On the other hand, vectors are ordered arrays of single numbers. These are examples of 1st-order tensors. Vectors are members of objects called vector spaces. 

Now, we will discuss the concepts of the scalar product of vectors and scalar and vector matrices that together fall in the definition of linear algebra. All these terms have different meanings however are related to each other in some or other ways. Before beginning with the detailed explanation of scalar and vector meaning, let us understand the meaning of some of the important terms in brief.

Matrix

The matrix given below shows a 22 matrix, the elements are denoted by a₁₁ = 1, a₁₂ = 2, a₂₁ = 3, a₂₂ = 4.

\[\begin{bmatrix}1&2 \\ 3&4 \end{bmatrix}\] , 

Usually, the letter A is used to represent a matrix, and to denote a particular element of a matrix, we use lower case letters, i.e. a_ij. Here, we mean the row, and j means the columns. E.g., in the matrix 2, the element is denoted by a₁₁ = 1, a₁₂ = 2, a₁₃ = 3, and it goes on. Whereas below matrix shows a 24 matrix.

\[A=\begin{bmatrix}1 &2  &  3&4 \\ 5 &6  &7  & 8\end{bmatrix}\]

Special Matrix

It’s another kind of matrix where the value of the matrix is 0. This matrix is represented as:

\[\begin{bmatrix}0&  0&0 \\ 0&0  &0 \\ 0&0  &0 \end{bmatrix}\]

Scalar Vector

A vector is also a kind of matrix but with either one row or one column. E.g. a matrix with 1 row and 3 columns or 3 rows or 1 column would be considered a vector.

\[\begin{bmatrix}x_{1}\\x_{2} \\ \vdots \\x_{m} \end{bmatrix}^{T}=\begin{bmatrix}x_{1}&x_{2}\ldots  &x_{m}\end{bmatrix}\]

The above matrix represents a vector with a₁ column and m rows.

Scalar Matrix

A scalar matrix is a diagonal matrix where the value of all the diagonal elements is the same. E.g.,

\[\begin{bmatrix}2&0  &0 \\0 &2  &0 \\0 &0  &2 \end{bmatrix}\]

Scalar and Vector Product

As we discussed earlier, scalar and vector matrices are different from each other. However, in this section, the meaning of scalar is a single element or a matrix with just one element. It could be any number. E.g. 4 is a scalar quantity.

Now let us understand how to multiply a scalar quantity with a vector matrix. Again, we will understand it using an example.

4 x \[\begin{bmatrix}1\\2 \\3 \end{bmatrix}=\begin{bmatrix}1\times 4\\2\times 4 \\3\times 4 \end{bmatrix}=\begin{bmatrix}4\\8 \\12 \end{bmatrix}\]

In the above-mentioned figure, we are given 4 as scalar quantity and a 3 x 1 matrix, i.e. a vector matrix with 3 rows and 1 column. Let us denote the scalar quantity with A and vector-matrix with B.

A = 4

B = 3 x 1 matrix

Multiplying a Vector By a Scalar

Now, to multiply the scalar quantity with a vector matrix to find the scalar and vector product of two vectors, all you have to do is multiply the scalar quantity with all the elements of the vector-matrix to get a new matrix as a result of their product. The new matrix formed would have the same number of rows and columns as that of the vector matrix.

Conclusion

Here, you have learned the following things:

1. How to find the scalar and vector product of two vectors.

2. How to find the scalar product of vectors.

3. Scalar and vector meaning.

4. Multiplying a vector by a scalar, and some of the important terminologies that are often used while dealing with these problems.

Having said, the examples mentioned above along with their respective figures and formulas work for every type of vector-matrix irrespective of the number of elements. When it comes to other kinds of matrices apart from Vector, we can easily expand them to find out their values by using a formula.