(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 \( \mathcal{F} \) be the vector space of continuous functions \( f : \mathbb{R} \to \mathbb{R} \) and let \( H \) be the subspace of \( \mathcal{F} \) spanned by \(\{ 1, \sin(t), \cos(t), \sin(t) \cos(t) \} \).
**(a)** Let \( K \) be the set of functions \( K = \{ a \sin(2t) + \cos(t) : a \in \mathbb{R} \} \). 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 \( L = \{ a \cos^2(t) : a \in \mathbb{R} \} \). Determine whether \( L \) is a vector subspace of \( H \) or not. Justify your answer.
**(c)** Consider \( \mathbb{R} \) as a real vector space (under the usual real number addition and multiplication). Let \( T : H \to \mathbb{R} \) be the function given by \( T(f) = \int_0^{\pi} f(t) \, dt \). Show that \( T \) is a linear transformation.
**(d)** Find a nonzero vector in ker(\( T \)). Justify your answer.
**(e)** Let \( D : H \to \mathcal{F} \) 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:
1. For an \( m \times n \) matrix \( A \) and any vector \( b \subset \mathbb{R}^m \), the set of solutions \( x \) of the equation \( Ax = b \) is always a vector subspace of \( \mathbb{R}^n \).
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