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
.

Question

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
.

BlackTom AI · Accepted Answer

We start by identifying what the standard matrix of a linear transformation T: R^3 -> R^3 looks like. Its columns are exactly T(e1), T(e2), and T(e3), where e1, e2, e3 are the standard basis vectors.

From the given data:
- T([1,0,0]) = [1,2,3]. Since [1,0,0] = e1, the first column is T(e1) = [1,2,3].
- T([1,1,0]) = [4,5,6]. But [1,1,0] = e1 + e2, so T(e1) + T(e2) = [4,5,6]. We already have T(e1) = [1,2,3], so T(e2) = [4,5,6] − [1,2,3] = [3,3,3]. Thus the second column is T(e2) = [3,3,3].
- T([1,1,1]) = [7,8,9]. Note that [1,1,1] = e1 + e2 + e3, so T(e1) + T(e2) + T(e3) = [7,8,9]. We know T(e1) = [1,2,3] and T(e2) = [3,3,3], whose sum is [4,5,6]. Therefore T(e3) = [7,8,9] − [4,5,6] = [3,3,3]. Hence the third column is T(e3) = [3,3,3].

Collecting the columns, the standard matrix is:
[ [1, 3, 3],
  [2, 3, 3],
  [3, 3, 3] ]

Now, evaluate the provided options:
Option 1 presents a matrix with columns [1,2,3], [4,5,6], [7,8,9], which would correspond to T(e1) = [1,2,3], T(e2) = [4,5,6], T(e3) = [7,8,9]. This contradicts the given T(e2) and T(e3) deduced above, so it does not match.
Option 2 shows a matrix with first row [1,1,1], second row [0,1,0], third row [0,0,1], which is not consistent with the computed columns at all.
Option 3 displays a matrix with first row [1,3,2], second row [2,3,1], third row [3,3,0], which again does not align with the derived column vectors.
Option 4 contains the matrix:
[ [1, 3, 3],
  [2, 3, 3],
  [3, 3, 3] ]
This matches exactly the columns T(e1) = [1,2,3], T(e2) = [3,3,3], T(e3) = [3,3,3] we computed. Therefore, option 4 is the one that aligns with the given information.

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 .单项选择题

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