The action of a linear transformation
𝑇
on a basis of
𝑅
3
is given.

Find
𝑇
(
[
1

2

3
]
)

Question

The action of a linear transformation 
𝑇
 on a basis of 
𝑅
3
 is given.

Find 
𝑇
(
[
1

2

3
]
)

BlackTom AI · Accepted Answer

Let's unpack the problem by listing the given data and what we need to find.
- The transformation T is defined on a basis of R^3 consisting of three vectors:
  v1 = [1, 0, 1]^T, with T(v1) = [0, 1]^T
  v2 = [0, -1, 1]^T, with T(v2) = [1, 0]^T
  v3 = [1, -1, 0]^T, with T(v3) = [0, 0]^T
- We are asked to compute T([1, 2, 3]^T).

First, express the vector [1, 2, 3]^T as a linear combination of the basis vectors v1, v2, v3:
Find scalars a, b, c such that a*v1 + b*v2 + c*v3 = [1, 2, 3]^T.
This gives the system from the component equations:
  x-component: a + 0*b + 1*c = 1  => a + c = 1 ...(1)
  y-component: 0*a + (-1)*b + (-1)*c = 2  => -(b + c) = 2 => b + c = -2 ...(2)
  z-component: 1*a + 1*b + 0*c = 3  => a + b = 3 ...(3)

Now solve the system. From (1), c = 1 - a. From (3), b = 3 - a. Substitute into (2): (3 - a) + (1 - a) = -2 -> 4 - 2a = -2 -> -2a = -6 -> a = 3. Then b = 3 - 3 = 0 and c = 1 - 3 = -2.
So [1, 2, 3]^T = 3*v1 + 0*v2 - 2*v3.

Apply T linearly: T([1, 2, 3]^T) = 3*T(v1) + 0*T(v2) - 2*T(v3).
Given T(v1) = [0, 1]^T and T(v3) = [0, 0]^T, this becomes
  T([1, 2, 3]^T) = 3*[0, 1]^T + 0*[1, 0]^T - 2*[0, 0]^T = [0, 3]^T.

Now evaluate the provided options:
- Option 1: [2; 5] does not match [0; 3], so this is incorrect.
- Option 2: [3; 0] does not match [0; 3], so this is incorrect.
- Option 3: [5; 2] does not match [0; 3], so this is incorrect.
- Option 4: [0; 3] matches exactly what we computed, so this is correct.

The action of a linear transformation 𝑇 on a basis of 𝑅 3 is given. Find 𝑇 ( [ 1 2 3 ] )单项选择题

类似问题

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