Linear Algebra - 4

Hi,

So, we have covered some stuff about vectors and their linear combinations, let us now take a look at matrices. Matrices are mathematical objects used to represent arrays of numbers in a neat way. They are mostly used to represent a system of linear transformations. Please note that these are brief overview of concepts and not an in-depth analysis containing proofs. As mentioned in Part-1 of these blogs, these are meant as "notes" for the Khan Academy videos. Please watch the videos for detailed explanation of these concepts.

For example, $$ a * \vec x + b * \vec y = \vec c  |  \vec a, \vec b \in R^2$$$$ a * (x_1, x2_) + b * (y_1, y_2) = (c_1, c_2)  |  \vec a, \vec b \in R^2$$

This can be represented using matrices as, and let AB be the product. 
 
$${\begin{pmatrix} x_1 & y_1  \\  x_2 & y_2 \end{pmatrix}*\begin{pmatrix} a \\ b \end{pmatrix}=AB}$$

Now the right hand side of the equation previous to the above can be represented as,$${\begin{pmatrix} c_1 \\  c_2 \end{pmatrix}=AB}$$

Thus, using matrices, we can neatly represent the above linear system of equations.Operations that can be performed on matrices are:
- Addition
- Subtraction
- Scalar Multiplication
- Matrix Multiplication

Null Space of  a Matrix:

Since we have seen briefly what a matrix is, let us define the null space of matrix. We can visualise a matrix as a set of row vectors or as a set of column vectors.

Here, we can visualize the matrix as consisting of vectors(x1, x2) and (y1, y2) or as (x1, y1) and (x2, y2) $${\begin{pmatrix} x_1 & y_1  \\  x_2 & y_2 \end{pmatrix}}$$

Now, the Null-Space of a matrix A is the set of all vectors of the matrix such that
$$A \vec X = \vec 0 $$
If the above condition is true, then vector X belongs to the null-space of matrix A.
 
It is interesting to note that if the column vectors of a matrix are linearly independent, then 0 vector is the only element of the null-space of the matrix. 

Column Space of  a Matrix:

The column space of a matrix is the set of all possible combinations of all the column vectors of the matrix. Hmm, linear combinations of vectors? Ring a bell?
Yes, it is the span. Thus, column space of a matrix is nothing but the span of all the column vectors of A.

Consider a matrix$${A=\begin{pmatrix} x_1 & y_1  \\  x_2 & y_2 \end{pmatrix}}$$
Thus, $$C(A) = span(x, y) $$

Note: if the column vectors of a matrix are linearly independent, then they can be the basis for the column space of A.