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(a) Let W be the subspace of R² that consists of all vectors of the form (a, –7a), where a is a real number. Find W+, the orthogonal complement of W. (Hint: think about a straight line that is perpendicular to the line y = -7x at the origin)

(a) Let W be the subspace of R² that consists of all vectors of the form (a, –7a), where a is a real number. Find W+, the orthogonal complement of W. (Hint: think about a straight line that is perpendicular to the line y = -7x at the origin)

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) Let W be the subspace of R? that consists of all vectors of the form (a, – 7a), where a is a real number. Find W+, the orthogonal complement of W. (Hint: think about a straight line that is perpendicular to the line y = -7x at the origin) | (b) Let F be the set of functions of the form f(x) A sin(x) + Bcos(2.x), where A, B are some real constants. Show that there must exist exactly one function f in F so that for any fe F, V/ ($(x) – arctan(x))²dx < (f(x) – arctan(x))²dx
Transcribed Image Text:(a) Let W be the subspace of R? that consists of all vectors of the form (a, – 7a), where a is a real number. Find W+, the orthogonal complement of W. (Hint: think about a straight line that is perpendicular to the line y = -7x at the origin) | (b) Let F be the set of functions of the form f(x) A sin(x) + Bcos(2.x), where A, B are some real constants. Show that there must exist exactly one function f in F so that for any fe F, V/ ($(x) – arctan(x))²dx < (f(x) – arctan(x))²dx
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