How to determine if some Linear Transformation exists I have these vectors below:

And i have this question about this below:
Is there a linear transformation L such that L (u1) = u2 and L (u3) = u4?
If yes:
How many such L are there? Enter the default matrix for such an image
L. If the answer is no: Explain
I know that a Linear Transformation you can add them togheter and you can also multiply them by a scalar but i'don't know here?
And another thing is that i searched about this and they told me about Linear Independent vectors or something??
 A: First of all, we could just define a transformation by saying $L(u_1) = u_2$ and $L(u_3)=u_4$. This is possible because $u_1$ and $u_3$ are linearly independent (which means the only way to obtain zero as a linear combination of $u_1$ and $u_3$ is by plugging in coefficients zero:
$$a u_1 + bu_3 = 0 \Rightarrow a=b=0)$$
Why is it possible to "just define the transformation as we want"? Because a system of linearly independent vectors is contained in a basis of the vector space in which we are working, that means we can add more vectors to $\{u_1, u_3\}$ (in this case exactly one more) to form a basis of $\mathbb R^3$. And a linear transformation is described by how it acts on the basis vectors because every vector $v \in \mathbb R^3$ is a unique linear combination of the three basis vectors.
So: yes, there is a transformation $L$ which satisfies your given conditions. And since we also have infinitely many choices for what the image of the third basis vector should be, there are infinitely many such transformations.
