Your book is discussing in what cases you can be given some input-output pairs for an unknown transformation $T$ and can figure out what $T$ has to be from these. That is, can you decide whether there are more than one choice of $T$ that would have those input-output pairs, or is there only one possible choice for $T$?
What that sentence is telling you is that, if you are given the outputs of $T$ for a collection of input vectors which forms a basis for the whole domain of $T$, then you know what $T$ must be on every vector in the domain. That is, you can figure out exactly what $T$ is, and it can only be one possible transformation. This follows from what it means to be a linear transformation.
If you're only given information about what $T$ does to some of the basis vectors but not all, then there are multiple choices for what $T$ could be, because you didn't get enough information. In that case, you can't figure out what $T$ is.
The examples are about whether we can figure out what the unknown linear transformation $T$ is, given a few of its values.
The first example looks like a linear transformation $T$ defined on the vector space of lines in the plane: $\{ax+b\mid a,b\in \mathbb{R}\}$, and mapping to points in the plane. You've been told what it does to two particular lines, $9x+5$ and $7x+4$. The relevant question here is: Is this enough information to decide what $T$ does to every line in the plane? Your book points out that this is enough information, because every line in the plane can be expressed as a linear combination of these two lines. That means that the two lines they gave you the value of $T$ on form a basis for the space consisting of all lines. Since $T$ is a linear transformation, it is determined by its value on these basis vectors.
The second example is a linear transformation that's going the other way: it sends points in the plane to lines in the plane. You're told the value for two points, and again, the relevant question is whether this is enough information to decide what $T$ does on every point. The book points out that it is not enough, because we can find two distinct linear transformations which have these same input-output pairs. That means that these input-output pairs do not uniquely identify the transformation $T$ that we're talking about. We can't figure out what $T$ is in this case. The reason is that the input points $(2,1)$ and $(6,3)$ don't form a basis for the plane.
As far as how the authors found the two possible transformations for the second example: I have an idea, and it is related to all of this, but I'm not sure how much you know yet. Read on if you like. They may have used that if a point $(a,b)$ is mapped by a linear transformation to a line $mx+c$, then $m$ has to be a linear combination of $a$ and $b$, and $c$ has to be a (possibly different) linear combination of $a$ and $b$. Using the constraints (the input-output pairs) we have, this lets us set up a system of equations. We must have that for some coefficients $c_1, c_2, c_3, c_4$:
1) $c_1(2) + c_2(1) = 4$
2) $c_3(2) + c_4(1) = 5$
3) $c_1(6) + c_2(3) = 12$
4) $c_3(6) + c_4(3) = 15$
This lets us set up an augmented matrix:
$$
\begin{pmatrix}
2 & 1 & 0 & 0 & 4 \\
0 & 0 & 2 & 1 & 5 \\
6 & 3 & 0 & 0 & 12 \\
0 & 0 & 6 & 3 & 15 \\
\end{pmatrix}
$$
You might notice that the 3rd row is a multiple of the first, and the fourth is a multiple of the second, so this reduces to a matrix with rank 2. That leaves 2 free parameters for us to vary, and these free parameters correspond to choices we have in coming up with possible transformations. The authors likely just selected some parameters. However, they might have just guess-and-checked as well.
In contrast, you would find that if you tried to repeat this matrix idea with the first example, you'd end up with a full rank matrix. There would be no free parameters. Staring at the above matrix for a while might give you some additional insight into what happens when the input values are or are not linearly independent.
The kernel of this map is the following:
$$ker(T) = \{ a_0 + a_1 x + a_2 x^2 + a_3 x^3 \in P_3 : a_3 x^2 - a_0 = 0\}$$
If this polynomial $a_3 x^2 - a_0$ is equal to zero for all $x$ values, then we know that $a_3$ and $a_0$ must be zero. So the kernel is in fact the set of all polynomials in $P_3$ with $a_3 = a_0 = 0$, in other words, all polynomials of the form $a_1 x + a_2 x^2$.
You can somewhat check this result using rank nullity. The dimension of $P_3$ is equal to $4$. The dimension of the image is $2$, and hence the dimension of the kernel is $4-2 = 2$.
Best Answer
I assume $P_3$ is the set of polynomials with real coefficients having degree $\leq3$.
How does the general element $p\in P_3$ look like? (Note that a polynomial, albeit being a complicated expression, is considered here as a "point" in $P_3$ and is denoted therefore by a single letter $p$.)
Verify: $P_3$ is a vector space.
Produce a basis of $P_3$, i.e., an array $\bigl(e_k)_{1\leq k\leq n}$ of special polynomials such that any $p\in P_3$ can be written as a linear combination (with real coefficients) of these $e_k$. What is $n$ ?
Verify that $T:\ P_3\to{\mathbb R}$ is linear, i.e., that for arbitrary $p$, $q\in P_3$ and arbitrary $\alpha$,$\beta\in{\mathbb R}$ one has $T(\alpha p+\beta q)=\alpha T(p)+\beta T(q)$.
Now we are proceeding to the analysis of $T$. To do this we have to find the matrix of $T$. How many rows resp. columns does this matrix have? To find the matrix we have to compute the effect of $T$ on the $n$ basis vectors and to write the result into the columns of the matrix. This means that we actually have to compute $n$ integrals.
This should do for the moment $\ldots$