91Ó°ÊÓ

A linear transformation is a function from one vector space to the other vector space that respects the linear structure of each vector space.

For example,

The defining characteristic of a linear transformation.

$T:V→W$Is that, for any vectors x and y in V and scalars and b of the underlying fields.

$T(ax+by)=aT\left(x\right)+bT\left(y\right)$

Represent an arbitrary polynomial of degree n as.

$p\left(t\right)=a_n{t}^n+{a_{n-1}}_{n-1}....+a_{1}t+a_0$

Let’s define linear transformation T as:

$T\left(p\right)=0.t^n+0.t^{n-1}+......+0.t^6+a_5{t}^5+....+a_{1}t+a_0$

Then T is a linear transformation from P to $P_5$whose image is all of$P_5$ because T maps every polynomial from $P_5$to itself and every polynomial that is from P and not from reduced to a polynomial of degree 5 or less.

Hence, the statement is true.