Let
𝑇
be the linear transformation defined as

𝑇
(
𝑥
)
=
𝐴
𝑥

where
𝐴
=
[
1

−
1

−
3

3
]
.

Then,
𝑇
(
𝑥
)
=
[
1

1
]
has

Question

Let 
𝑇
 be the linear transformation defined as

𝑇
(
𝑥
)
=
𝐴
𝑥

where 
𝐴
=
[
1
	
−
1

−
3
	
3
]
.

Then, 
𝑇
(
𝑥
)
=
[
1

1
]
 has

BlackTom AI · Accepted Answer

Let me restate the problem to ensure we’re examining the same thing: we want to solve the matrix equation A x = b where A = [[1, -1], [-3, 3]] and b = [1, 1]^T. The question then asks which of the provided options describes the set of solutions to T(x) = [1, 1]^T, with T(x) = A x.

First, observe the structure of A. The second row [-3, 3] is exactly -3 times the first row [1, -1]. This means the two rows are linearly dependent, so the row space (and thus the column space of A) has dimension 1. In practical terms, the two equations given by Ax = b cannot be independent: the system is consistent only if the second equation is a dependent copy of the first, scaled in the same way as the corresponding components of b.

Let’s write the system explicitly: 
Equation 1: x1 − x2 = b1,
Equation 2: −3 x1 + 3 x2 = b2.
Because Equation 2 is simply −3 times Equation 1 in the left-hand side, any solution must satisfy b2 = −3 b1 as well for the system to be consistent.

Now plug in b = [1, 1]^T. Here b1 = 1 and b2 = 1. Check the necessary consistency condition: is 1 equal to −3 times 1? No, 1 ≠ −3. Therefore the second equation cannot be satisfied simultaneously with the first equation for any x. The system is inconsistent.

Option-by-option analysis:
- No solutions: This matches the inconsistency identified above. Since the two equations are incompatible given b, there is no x that satisfies both equations at once.
- Infinitely many solutions: This would require the system to be consistent with at least one free variable, typically arising when the rank of A equals the rank of the augmented matrix [A|b] and is less than the number of unknowns. Here the inconsistency prevents any solution, so this option is incorrect.
- A unique solution: A unique solution would occur if A were full rank (rank 2) and the right-hand side b lay in the column space. However, as noted, the rows are dependent, so A is rank-deficient (rank 1). Given the inconsistent b, there is no single x that solves the system, making this option also incorrect.

Additional intuition: because the second equation is just a multiple of the first on the left, the only way Ax = b could hold is if the same multiple held for the right-hand side as well. Since b components do not share that multiple relationship, the equations clash instead of reinforcing each other, leading to no solutions.

In summary, examining the dependence in A and comparing b’s components shows the system cannot be satisfied by any x. The reasoning hinges on the proportionality between the rows of A and the mismatch with b’s components, which rules out any solution and rules out both infinite and unique solutions.

Let 𝑇 be the linear transformation defined as 𝑇 ( 𝑥 ) = 𝐴 𝑥 where 𝐴 = [ 1 − 1 − 3 3 ] . Then, 𝑇 ( 𝑥 ) = [ 1 1 ] has 单项选择题

类似问题

Which ones of the following linear transformations are NOT invertible? (Choose one or more)

Which ones of the following linear transformations are NOT invertible? (Choose one or more)

Which ones of the following linear transformations are NOT invertible? (Choose one or more)

What kind of transformation is performed by the matrix 𝐴 = [ 1 0 0 − 1 ] ?

Which of the following transformations 𝑇 : 𝑅 2 → 𝑅 2 is not linear?

Let 𝑇 : 𝑅 3 → 𝑅 3 be a linear transformation and suppose that 𝑇 ( [ 1 0 0 ] ) = [ 1 2 3 ] , 𝑇 ( [ 1 1 0 ] ) = [ 4 5 6 ] , 𝑇 ( [ 1 1 1 ] ) = [ 7 8 9 ] Find the standard matrix for the transformation 𝑇 .

The action of a linear transformation 𝑇 on a basis of 𝑅 2 is given. Find the matrix of 𝑇 .

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

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单