Consider the vectors ā₁ 8. , ā₂ = = (a) Write b as a linear combination of a₁, 72, and a3. (b) Suppose that T: R³ R2 is a linear transformation such that = *(E)--(E) --- () -- T = T = and T
Consider the vectors ā₁ 8. , ā₂ = = (a) Write b as a linear combination of a₁, 72, and a3. (b) Suppose that T: R³ R2 is a linear transformation such that = *(E)--(E) --- () -- T = T = and T
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Consider the vectors
ā₁ =
8. - [3]
, ā₂ =
, az =
b=
(a) Write b as a linear combination of a1, a2, and a3.
(b) Suppose that T: R³ → R² is a linear transformation such that
T
(E)-- () ---- () -
=
T
=
and T
1].
Is there a vector € R³ such that T(T) = 6? If not, explain why not; if so, give an example of such a
vector 7.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa923b6f-81dd-482c-8885-6de6bc295751%2F72ab7d62-4134-4424-a968-2d5f51d722d1%2F0n8qpu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the vectors
ā₁ =
8. - [3]
, ā₂ =
, az =
b=
(a) Write b as a linear combination of a1, a2, and a3.
(b) Suppose that T: R³ → R² is a linear transformation such that
T
(E)-- () ---- () -
=
T
=
and T
1].
Is there a vector € R³ such that T(T) = 6? If not, explain why not; if so, give an example of such a
vector 7.
Transcribed Image Text:Answer
For part (a), there are infinitely many ways to write b as a linear combination of a₁, a2, and a3. One
possibility is -3a₁ + 2a2 + a3 = b. For part (b), the answer is yes, and there are infinitely many such 7.
-3
One such is
2
using part (a) and the linearity of T.
1
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