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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
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.
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.
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
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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