Question

Let:T : ℝ3→ℝ3 be the transformation that projects each vector x=​(x1​,x2​,x3​)onto the plane

Answer 1

Answer:

Step-by-step explanation:

A linear transformation must satisfy the following properties.

- T(0) = 0.

- For vector a,b then T(a+b) = T(a) + T(b).

- For a vector a and a scalar r, it must happen that T(ra) = rT(a)

In this case we have that T(a,b,c) = (a,0,c).

Note that T(0) = T(0,0,0) = (0,0,0) = 0. So, the first property holds.

Let $a=(a_1,a_2,a_3), b=(b_1,b_2,b_3)$. Then

$T(a+b) = T((a_1+b_1,a_2+b_2,a_3+b_3)) = (a_1+b_1,0,a_3+b_3) = (a_1,0,a_3)+(b_1,0,b_3) = T(a) + T(b)$

So the second property holds.

Finally, let r be a scalar and let $a=(a_1,a_2,a_3)$. Then

$T(ra) = T((ra_1,ra_2,ra_3)) = (ra_1,0,ra_3) = r(a_1,0,a_3)= rT(a)$

So, the three properties hold, and therefore, T is a linear transformation.