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Let L be defined by the vector equation x =~t and T : R2 R2 be a linear -1transformation T : x+ Ax with A =Find T(L)....

Question

Let L be defined by the vector equation x =~t and T : R2 R2 be a linear -1transformation T : x+ Ax with A =Find T(L).

Let L be defined by the vector equation x = ~t and T : R2 R2 be a linear -1 transformation T : x+ Ax with A = Find T(L).



Answers

Let $T : \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ be the transformation that projects each vector $\mathbf{x}=\left(x_{1}, x_{2}, x_{3}\right)$ onto the plane $x_{2}=0,$ so $T(\mathbf{x})=$ $\left(x_{1}, 0, x_{3}\right) .$ Show that $T$ is a linear transformation.

Okay, So for the ocean transformation t is are three on two or three. Okay. And it would be the transformation that reflects each vector X and the vector X is given x one x two and x three. Okay, through the plane. Extra equals to zero and on to And the images images given, that is X one x two and minus X three. Okay, we have show that is a linear transformation. Okay, So considered you be in R three with this killer city. Okay, We are considering UV in our three with his killers. See? And the and lead and let X equals to see you plus Devi. Okay, so now it will be we can write that X equals toe. See you one plus d We even okay and see you two plus D V two and c u three plus d V three. Okay, And now the X because toe TCU plus D v Then we can write this. See you one plus d V one. See you two plus d. We do and transformation will be minus then minus See you three plus the eatery. Okay. So further solving that we will get to see you One plus Devi even see you took less DV two and minus See you three minus D v three. Okay, so far that solving their distance Formacion image way get See you one See you do and minus See you three glass Devi even DV two on minus day with three Okay, so we will put Get see Comin out of this This will be you and you two and minus you three plus be comin out off their that we won we do on minus with three So it will be See do you class d TV? So this is a Christian off linear transformation Okay, so we can say that you can say that he is a linear transformation. Thank you. This is our final unset. Thank you.

So the Sol this suit solve this problem? The first thing I would like to know is that the transformations are linear, which means I can just shooting the tea over to to be one distributed over 23 v to assume Iraq to distribute to your the one and distribute over B two. So our expression to come Teoh to be one musty of three V two, which is able to be Willen plus B two. In a similar vein, it's equal to the want TV one post here, the to which is equal to three, the one minus V two and then secondly, because he is later, I can pull out my cost, a term, the two and the cost term of three. So this expression becomes two times T O. V. One because three times t o V two, which is unequal to be one plus it be to So in order, stall for the system of equations. I'm going to subtract from this equation from this one by multiplying it by two. So it goes like this. So it's a fact. Two times t of the one plus trv to equal to minus two times three V one minus B two. So this expression becomes TRV to is equal to the one plus V two minus two times three v one minus feature. So what TV to it. One with simple I it comes, is equal to minus four b one plus three be to so now defined TRV one. We just take one of our equations and substitute to the two into it, the simplest one being the second equation. So it turns out TV one plus TV too equals Ah three V one. Mine isn't you to substituting for to be too. We end up with TLV one plus minus four. Do you want plus three? The two is equal to three. B one amounted to be to getting TV one by itself into collecting like terms we end up with TV one is equal to 71 minus four to be to


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