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Let:T : ℝ3→ℝ3 be the transformation that projects each vector x=​(x1​,x2​,x3​)onto the plane
x2=0​,
so
​T(x​)=​(x1​ ,​0 ,x3​). Show that T is a linear transformation.
The first property for T to be linear is
​T(0​) =_________________________________________________
Check if this property is satisfied for T.T​(x1​, x2​, x3​) =​(x1​, ​0,x3​)
​T(0,0,0) = ( _____ ,_____, _,_______) ,
​So, is the first property​ satisfied?
Yes 0r No
The second property for T to be linear is ​T(cu​+dv​)=______(see choices below)___________________________________________for all vectors(u and v)i n the domain of T and all scalars​ c, d
cT(u)+dT(v)
dT(u)+cT(v)dT(u)−cT(v)
cT(u)−dT(v)
for all vectors
u​,
v in the domain of T and all scalars​ c, d.
Check if this property is satisfied for T. Let u=​(u1​, u2​, u3​) and v =​(v1​, v2​, v3​).
T(cu+dv)=​(cu1+dv1​, ​0, cu3+dv3​) =​(cu1​, _______, _________) +​(dv1​, ________, _________)
Factor out the scalar in each ordered triple.
T(cu+dv) = ____​(u1​, ​0, u3) + ______​(v1​, ​0,v3​)
Further simplify the previous equation.
T(cu+dv) = c
▼
(choices pick one)
T(v)
T(u)+d
▼
(choices for the arrow above pick one)
T(v)
T(u)
​So, is the second property​ satisfied?
Yes
No
​Thus, T ▼
is
is not
linear.

Answer :

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 [tex] a=(a_1,a_2,a_3), b=(b_1,b_2,b_3) [/tex]. Then

[tex]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)[/tex]

So the second property holds.

Finally, let r be a scalar and let [tex] a=(a_1,a_2,a_3)[/tex]. Then

[tex] T(ra) = T((ra_1,ra_2,ra_3)) = (ra_1,0,ra_3) = r(a_1,0,a_3)= rT(a)[/tex]

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

MrRoyal

It is true that T represents a linear transformation

How to determine if T is a linear transformation

A linear transformation is such that have the following properties

  1. T(0) = 0.
  2. If u and v are vectors, then T(u+v) = T(u) + T(v).
  3. If u is a vector and r is a scalar, then T(ru) = rT(u)

For the first property, we have:

T(x1,x2,x3) = (x1,0,x3)

The above property becomes

T(0) = T(0,0,0)

T(0) = (0,0,0)

T(0)= 0.

So, we can conclude that the first property of the linear transformation is satisfied

For the second property, we make use of the following:

u = (u1, u2, u3) and b = (v1,v2,v3)

The above property becomes

T(u + v) = T(u1 + v1, u2 + v2, u3 + v3)

Expand

T(u + v) = T(u1 + v1, 0, u3 + v3)

Expand

T(u + v) = (u1,0,u3) + (v1, 0, v3)

Simplify

T(u + v) = T(u) + T(v)

The above means that the second property is also satisfied

Recall that:

u = (u1, u2, u3)

So, we have:

T(ru) = (ru1, ru2, ru3)

Where r is a scalar

Expand

T(ru) = (ru1, 0, ru3)

Further, expand

T(ru) = r(u1, 0, u3)

So, we have:

T(ru) = rT(u)

The above means that, the third property is also satisfied

Hence, T represents a linear transformation

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