Determining linear transformation

linear algebralinear-transformations

Let $V$ be a vector space, and $T:V→V$ a linear transformation such that:

$T(2v_1 + 3v_2) = -5v_1 – 4v_2$ and $T(3v_1 + 5v_2) = 3v_1 -2v_2$

Then:

T(v1)= ? v1+ ? v2

T(v2)= ? v1+ ? v2

T(4v1+2v2)= ? v1+ ? v2

I cannot solve this problem and have been at it for hours. I found a similar question here: Finding the basis of a vector space. I tried applying the same operations, but do not understand how they got to the final solution.

Best Answer

Simple rule of linear transformation,

$T(c \alpha+d \beta)=c T(\alpha)+d T(\beta)$, where $c,d$ are scalars.

Take $\alpha=2 v_1 +3 v_2 , \beta=3 v_1 + 5 v_2 $

Now , you have to figure out for what values of $c,d$ ,
$c \alpha + d \beta = v_1 $ for finding the value $T(v_1)$

Now , from $c \alpha + d \beta = v_1 $ ,

We get,

$(2c+3d) v_1 + (3c+5d) v_2 = v_1 $

So, $2c+3d=1$ and $3c+5d=0 $

Solving we get, $c=5 , d=-3$

So, $T(5 \alpha - 3 \beta) = 5 T(\alpha)- 3 T(\beta) $

$\implies$ $ T(v_1)= $ ?

Now , I think you can figure out the rest.

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