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Determine, with justification whether each of following sets forms a linear space? (i) The set of quadratic polynomials {p: p(x) = ax²+bx+c, a ‡0}, (ii) the line L in R³ given by {(1,2,3)+t(2,4,6), t ≤ R}, iii) the nxn invertible real matrices, (iv) the set of functions {f: R→R: there exist A, B E R such that f(x) = Acosx+Bsinx and f(3) = 0}.

Determine, with justification whether each of following sets forms a linear space? (i) The set of quadratic polynomials {p: p(x) = ax²+bx+c, a ‡0}, (ii) the line L in R³ given by {(1,2,3)+t(2,4,6), t ≤ R}, iii) the nxn invertible real matrices, (iv) the set of functions {f: R→R: there exist A, B E R such that f(x) = Acosx+Bsinx and f(3) = 0}.

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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(a) Determine, with justification whether each of following sets forms a linear space? (i) The set of quadratic polynomials {p: p(x) = ax²+bx+c, a ‡0}, (ii) the line L in R³ given by {(1,2,3)+t(2, 4, 6), t ≤ R}, (iii) the nxn invertible real matrices, (iv) the set of functions {f: R→R: there exist A, B E R such that f(x) = Acosx +Bsinx and f(3) = 0}. (b) A basis of a linear space E is a set of vectors which is linearly independent and spanning. Define what is meant by the terms (i) linearly independent and (ii) spanning. (c) Consider the vectors v₁ = (2,3,-1) and v₂ = (4,4,1) in R³. (i) Show that v₁ and v2 are linearly independent. (ii) Show that v₁ and v₂ do not span R³. (iii) Find a vector v3 such that {V₁, V2, V3} forms a basis of R³. (iv) Find an orthogonal basis of R³ that includes v₁.
Transcribed Image Text:(a) Determine, with justification whether each of following sets forms a linear space? (i) The set of quadratic polynomials {p: p(x) = ax²+bx+c, a ‡0}, (ii) the line L in R³ given by {(1,2,3)+t(2, 4, 6), t ≤ R}, (iii) the nxn invertible real matrices, (iv) the set of functions {f: R→R: there exist A, B E R such that f(x) = Acosx +Bsinx and f(3) = 0}. (b) A basis of a linear space E is a set of vectors which is linearly independent and spanning. Define what is meant by the terms (i) linearly independent and (ii) spanning. (c) Consider the vectors v₁ = (2,3,-1) and v₂ = (4,4,1) in R³. (i) Show that v₁ and v2 are linearly independent. (ii) Show that v₁ and v₂ do not span R³. (iii) Find a vector v3 such that {V₁, V2, V3} forms a basis of R³. (iv) Find an orthogonal basis of R³ that includes v₁.
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