Prove that the given set B is a basis for the given vector space V. And then, determine if the given vector is in V. If so, find the coordinate vector of ✔ with respect to B. a) B = 0 2 { (²9) · (₂2)⋅ ( ) ( ) }· 2 1 1 V = M₂2x2 (R), v = 5 (₁2) 1 -1 2 b) 8 = {(1, 2) - ( 7² )} V - Spa(8), (1²3) B= = v = 0 -1 1 1 0 2 (c) B = {1, 1+ 2X, 1+ 2X +3X², 1+ 2X + 3X² + 4X³}, V = R³(X), v = 8X³
Prove that the given set B is a basis for the given vector space V. And then, determine if the given vector is in V. If so, find the coordinate vector of ✔ with respect to B. a) B = 0 2 { (²9) · (₂2)⋅ ( ) ( ) }· 2 1 1 V = M₂2x2 (R), v = 5 (₁2) 1 -1 2 b) 8 = {(1, 2) - ( 7² )} V - Spa(8), (1²3) B= = v = 0 -1 1 1 0 2 (c) B = {1, 1+ 2X, 1+ 2X +3X², 1+ 2X + 3X² + 4X³}, V = R³(X), v = 8X³
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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solve parts a, b, and c
Transcribed Image Text:2. Prove that the given set B is a basis for the given vector space V. And then, determine if
the given vector v is in V. If so, find the coordinate vector of v with respect to B.
(a) B =
{ ( ) ( ))⋅ ( ² ) · ( ) }
"
(b) B =
V = M2x2 (R), V =
1
-1
{ ( 1 ) · G
9) ·( 1 )}
-1
(c) B = {1, 1+ 2X, 1 + 2X +3X², 1+ 2X +3X² + 4X³}, V = R3(X), v = 8X³
5
(₁22)
-2
V = Span(B), v =
(1 2 3)
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Step 1: Analysis and Introduction:
VIEWStep 2: Part a) - Prove they form the basis
VIEWStep 3: Part a) - Find the Coordinate of the vector.
VIEWStep 4: Part b) - Prove they form the basis
VIEWStep 5: Part b) - Find the Coordinate of the vector.
VIEWStep 6: Part c) - Prove they form the basis
VIEWStep 7: Part c) - Find the Coordinate of the vector.
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