Let V denote the vector space of all functions f : R → R, with the usual definitions of addition and scalar multiplication. Consider f, g e V such that f(x) = x and g(x) = e* for all x E R. Let S = {f,g} and W = span(S). Which one of the following statements is true? O a. S is a linearly independent subspace of V. O b. S is a spanning set for V and a basis for W O c. S spans V but is not linearly independent. O d. Sis a linearly independent subset of V and a basis for W
Let V denote the vector space of all functions f : R → R, with the usual definitions of addition and scalar multiplication. Consider f, g e V such that f(x) = x and g(x) = e* for all x E R. Let S = {f,g} and W = span(S). Which one of the following statements is true? O a. S is a linearly independent subspace of V. O b. S is a spanning set for V and a basis for W O c. S spans V but is not linearly independent. O d. Sis a linearly independent subset of V and a basis for W
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:Let V denote the vector space of all functions f : R
R, with the usual definitions of addition and scalar multiplication.
Consider f, g € V such that f(x)
= x and g(x) = e* for all x E R. Let S = {f,g} and W =
span(S).
Which one of the following statements is true?
S is a linearly independent subspace of V.
а.
b. S is a spanning set for V and a basis for W
O c. S spans V but is not linearly independent.
O d. S is a linearly independent subset of V and a basis for W
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