(a) Let V be a vector space and v₁, ..., V₂ € V. When do we say that vectors V₁, ..., Vn span V? (Give a precise definition.) (b) Consider vectors V1 = 1 2 V2 = 2 3 4 , V3 = 5 and v4= 8 (6) 10 in R³. (i) Do vectors V₁, V2, V3, V4 Span R³? (ii) Do vectors v₁ and v₂ span R³? (iii) Are vectors V₁, V2, V3, V4 linearly independent? Justify your answer in each case, and state precisely any theorems you use.
(a) Let V be a vector space and v₁, ..., V₂ € V. When do we say that vectors V₁, ..., Vn span V? (Give a precise definition.) (b) Consider vectors V1 = 1 2 V2 = 2 3 4 , V3 = 5 and v4= 8 (6) 10 in R³. (i) Do vectors V₁, V2, V3, V4 Span R³? (ii) Do vectors v₁ and v₂ span R³? (iii) Are vectors V₁, V2, V3, V4 linearly independent? Justify your answer in each case, and state precisely any theorems you use.
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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Transcribed Image Text:(a) Let V be a vector space and v₁, ..., Vn E V. When do we say that
vectors V₁, ..., Vň span V?
n
(Give a precise definition.)
(b) Consider vectors
V₁ =
V1
2
(1) ... - (1)
V2 = 3
"
V3 =
5
8
(-)
6 and v4= 9
(10)
in R³
(i) Do vectors V₁, V2, V3, V4 span R³?
(ii) Do vectors v₁ and v₂ span R³?
(iii) Are vectors V₁, V2, V3, V4 linearly independent?
Justify your answer in each case, and state precisely any theorems you use.
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