(Vector spaces) Let F be the vector space of continuous functions f: R → R and let H be the subspace of F spanned by {1, sin(t), cos(t), sin(t) cos(t)}. (a) Let K be the set of functions K = {a sin(2t) + cos(t): a ER}. Of the three properties of subspaces, which does K satisfy? Is K a vector subspace of H or not? Justify your answer. (b) Let L be the set of functions L = {a cos² (t): a ER}. Determine whether L is a vector subspace of H or not. Justify your answer. (c) Consider R as a real vector space (under the usual real number addition and multiplica- tion). Let T: HR be the function given by T(f) = f f(t) dt. Show that T is a linear transformation. (d) Find a nonzero vector in ker(T). Justify your answer. (e) Let D: HF be given by D(f) = f', the first derivative. Show that D is a linear transformation. (f) Find a spanning set for ker(D), where D is the linear transformation from part (c). Justify your answer. (g) Answer TRUE, FALSE or NONSENSE:- i. For an m x n matrix A'and any vector b CR, the set of solutions x of the equation Ax = b is always a vector subspace of R". ii. Every linear transformation is the null space of some vector space. iii. If A is an m x n matrix, then Null(4) is a vector subspace of R". iv. An m x n matrix A is linearly independent if it spans R
(Vector spaces) Let F be the vector space of continuous functions f: R → R and let H be the subspace of F spanned by {1, sin(t), cos(t), sin(t) cos(t)}. (a) Let K be the set of functions K = {a sin(2t) + cos(t): a ER}. Of the three properties of subspaces, which does K satisfy? Is K a vector subspace of H or not? Justify your answer. (b) Let L be the set of functions L = {a cos² (t): a ER}. Determine whether L is a vector subspace of H or not. Justify your answer. (c) Consider R as a real vector space (under the usual real number addition and multiplica- tion). Let T: HR be the function given by T(f) = f f(t) dt. Show that T is a linear transformation. (d) Find a nonzero vector in ker(T). Justify your answer. (e) Let D: HF be given by D(f) = f', the first derivative. Show that D is a linear transformation. (f) Find a spanning set for ker(D), where D is the linear transformation from part (c). Justify your answer. (g) Answer TRUE, FALSE or NONSENSE:- i. For an m x n matrix A'and any vector b CR, the set of solutions x of the equation Ax = b is always a vector subspace of R". ii. Every linear transformation is the null space of some vector space. iii. If A is an m x n matrix, then Null(4) is a vector subspace of R". iv. An m x n matrix A is linearly independent if it spans R
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:(Vector spaces) Let F be the vector space of continuous functions f: R → R and let H be the
subspace of F spanned by {1, sin(t), cos(t), sin(t) cos(t)}.
(a) Let K be the set of functions K = {a sin(2t) + cos(t): a € R}. Of the three properties
of subspaces, which docs K satisfy? Is K a vector subspace of H or not? Justify your
answer.
(b) Let L be the set of functions L
mm
{a cos²(t): a ≤ R}. Determine whether L is a vector
subspace of H or not. Justify your answer.
(c) Consider R as a real vector space (under the usual real number addition and multiplica-
tion). Let T: H →→ R be the function given by T(f) = f f(t) dt. Show that I is a linear
transformation.
(d) Find a nonzero vector in ker(T). Justify your answer.
(e) Let D: HF be given by D(f) = f', the first derivative. Show that D is a linear
transformation.
(f) Find a spanning set for ker(D), where D is the linear transformation from part (e). Justify
your answer.
(g) Answer TRUE, FALSE or NONSENSE: -
i. For an m x n matrix A'and any vector b CR, the set of solutions x of the equation
Axb is always a vector subspace of R".
ii. Every linear transformation is the null space of some vector space.
iii. If A is an m x n matrix, then Null(A) is a vector subspace of R.
iv. An m x n matrix A is linearly independent if it spans R
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