I am having problems understanding how to solve this question.
Find a linear function that satisfies both of the given conditions.
$f(-1) = 5, f(1) = 6$
Thanks,
Note: i have the answer, just need help understanding
[Math] linear function that satisfies both conditions
functionslinear algebra
Related Solutions
I will go through it very slowly so that you have lots of examples of how do things like this. Please don't be insulted; it just pays to be careful because this is a very detail-oriented task. First give the things you want to find names: they're natural numbers so I'll call one $n$, and one $k$.
Then look at the information, piece by tiny piece:
The sum of …
$$+$$
… two nonnegative numbers …
$$n+k$$ $$(n\geq 0, k\geq 0)$$
… is …
$$n+k=$$ $$(n\geq 0, k\geq 0)$$
… $1$.
$$n+k=1.$$ $$(n\geq 0, k\geq 0)$$
That wasn't so bad! The second sentence is yet another bunch of words we want to turn into symbols, so here goes:
The sum of …
$$+$$
… the square of …
$$()^2+$$
… one [number] …
Now we have a choice; we can either use $n$ or $k$. I'll choose $n$:
$$(n)^2+$$
… and twice …
$$(n)^2+(2*)$$
… the square of …
$$(n)^2+(2*()^2)$$
… the other [number] …
This time we do not have a choice; we must use $k$ (or if we had used $k$ last time, then this time we must use $n$): $$(n)^2+(2*(k)^2)$$
… [Express this] as …
$$(n)^2+(2*(k)^2)=$$
… a function of …
$$(n)^2+(2*(k)^2) = f()$$
… one of the numbers.
Again, we have a choice; I'll choose $k$ this time.
$$(n)^2+(2*(k)^2) = f(k)$$
In those last few steps, we took the final form, which without parentheses is $n^2+2k^2$, and wrote it in terms of either $n$ or $k$, using function notation. From the first part of the problem we have a relationship between $n$ and $k$; from here it is a matter of algebraic manipulation.
As a rule of thumb for turning text into symbols, you should have written down at least one new piece of information every time the word "of" appears. And usually it appears quite often. Every time a pronoun appears (especially the "missable" ones like "one" and "other" and sometimes "that") you should hunt to figure out what thing it's referencing; it is probably something else to write down.
[[I think a very reasonable question is how I knew what to do when I got to "as". I can't really say why I knew that this was the end of the "Express… as" clause. However, had I not put the equals sign there, I would have immediately known I had done something wrong, because when the "function" came up, there would be nowhere in the expression to put it.
So then I would have to read it carefully again until I realized that was because I needed the equals sign. This seems like a more practical approach: if you're not sure where something goes, it might be because you missed a thing; reread it.]]
Let $u=(x_1,y_1,z_1)$ and $v=(x_2,y_2,z_2)$.
Can you then show $T(u+v)=T(u)+T(v)$?
Best Answer
Hint: What is the general form of a linear function?
($f(x) =$ something something $x$ something something...)