Question

Show that T(x1,x2, x3,x4)= 3x1 -7x2+5x4 is liner transformation by finding the matrix for transformation. Then find a basis for the null space of the transformation

Answer 1

A linear transformation is a function from one vector space to another that respects the underlying (linear) structure of each vector space.

An example of a linear transformation is the multiplication of matrices with each other, as in the above task. The initial matrix is four-dimensional identity matrix which is represented as (x1,x2, x3,x4) and can be shown as:

$$I= \begin{pmatrix} 1 & 0 &0&0\\ 0 & 1 &0&0\\ 0 & 0 &1&0\\ 0 & 0 &0&1\\ \end{pmatrix}$$

And. to make transformation T(x1,x2, x3,x4) and get 3x1 -7x2+5x4, we should multiply this matrix by matrix A:

$$A= \begin{pmatrix} 3 & 0 &0&0\\ 0 & -7 &0&0\\ 0 & 0 &0&0\\ 0 & 0 &0&5\\ \end{pmatrix}$$

And, using the property of the identity matrix, which when is multiplied by a matrix gives us the same matrix, we get:

$$I*A=A=I∗A=A= (3x1 -7x2+5x4)$$

So the operator T is multiplier by the matrix A and it is the linear transformator.

And then we should find a basis for the null space of the transformation. For this we should equate every line of A to 0 and find every x.

$$3x1=0$$

$$-7x2=0$$

$$5x4=0$$

And x1=x2=x4=0 gives us the right system of equations. x3 can be any real number, because it is multiplied by zero in matrix A and don't influence to the resulting matrix, it is simply absent in the answer. So we have a system of vectors in the basis:

$$\langle x1, x2, x3, x4 \rangle = \langle 0, 0, R, 0 \rangle$$

Where R is any number in the set of real numbers.