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Solutions Section 3.2 Fixed

The document provides solutions to exercises determining whether various sets form subspaces in different mathematical contexts, including R^2, R^3, R^(2x2), polynomial sets in P_4, and function sets. The responses indicate which sets qualify as subspaces, with specific answers for each case. Additionally, it confirms that C^1[a, b] and the set S = {B | AB = BA} are subspaces.

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35 views2 pages

Solutions Section 3.2 Fixed

The document provides solutions to exercises determining whether various sets form subspaces in different mathematical contexts, including R^2, R^3, R^(2x2), polynomial sets in P_4, and function sets. The responses indicate which sets qualify as subspaces, with specific answers for each case. Additionally, it confirms that C^1[a, b] and the set S = {B | AB = BA} are subspaces.

Uploaded by

sivakamaraj.klr
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© © All Rights Reserved
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Solutions to Section 3.

2 Exercises

1. Determine whether the following sets form subspaces of R^2:


(a) Yes, it is a subspace.
(b) No, it is not a subspace.
(c) Yes, it is a subspace.
(d) No, it is not a subspace.

2. Determine whether the following sets form subspaces of R^3:


(a) No, it is not a subspace.
(b) Yes, it is a subspace.
(c) Yes, it is a subspace.
(d) No, it is not a subspace.

3. Determine whether the following are subspaces of R^(2x2):


(a) Yes, it is a subspace.
(b) Yes, it is a subspace.
(c) Yes, it is a subspace.
(d) No, it is not a subspace.
(e) Yes, it is a subspace.
(f) Yes, it is a subspace.
(g) No, it is not a subspace.

4. Null space of matrices (see detailed solutions).

5. Determine whether polynomial sets form subspaces of P_4:


(a) Yes, it is a subspace.
(b) No, it is not a subspace.
(c) Yes, it is a subspace.
(d) No, it is not a subspace.

6. Determine whether function sets form subspaces:


(a) Yes, it is a subspace.
(b) Yes, it is a subspace.
(c) No, it is not a subspace.
(d) Yes, it is a subspace.
(e) No, it is not a subspace.

7. C^1[a, b] is a subspace of C[a, b] -> Yes.

8. S = {B | AB = BA} is a subspace of R^(n x n) -> Yes.

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