Help finding linear transformation Equation T:V -> V

linear algebralinear-transformations

Let $V$ be a vector space, and $$T:V→V$$ a linear transformation such that $$T(2v_1−3v_2)=−3v_ 1+2v_ 2$$ and $$T(−3v_ 1+5v_2)=5v_ 1+4v_2$$

Solve $$T(v_1), T(v_2), T(−4v_ 1−2v_2)$$

I tried to put it into an Augmented matrix, take the inverse, and apply the inverse in order to work backwards from there but I still don't have any luck.. Does anyone have tips?

Best Answer

You list some very overpowered things that you don't need for this problem. Remember that linearity is a very strong condition.

Hint: how do you "make" $v_1$ out of copies (linear combinations) of $2v_1-3v_2$ and $-3v_1+5v_2$?

Multiply the first thing by $5$, and the second thing by $3$

$5(2, -3) + 3(-3, 5) = (1, 0)$

$T(v_1) = T(5(2v_1-3v_2)+3(-3v_1+5v_2))= 5(T(2v_1-3v_2))+3(T(-3v_1+5v_2))=5(-3v_1+2v_2)+3(5v_1+4v_2)$ where the last inequality comes from your definition of $T$, and linearity of $T$

Now, can you do the other problems?

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