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9. Let V be a vector space. Decide if the following statements are true or false. Justify your claims. (a) The empty set is lincarly independent. (b) Any subset of a linearly independent set is linearly independent. (c) Any superset of a lincarly dependent set is linearly dependent.

9. Let V be a vector space. Decide if the following statements are true or false. Justify your claims. (a) The empty set is lincarly independent. (b) Any subset of a linearly independent set is linearly independent. (c) Any superset of a lincarly dependent set is linearly dependent.

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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9. Let V be a vector space. Decide if the following statements are true or false. Justify your claims. (a) The empty set is lincarly independent. (b) Any subset of a linearly independent set is linearly independent. (c) Any superset of a linearly dependent set is linearly dependent. (d) The zero set is linearly independent. (e) Any one-element set is linearly independent. (f) The union of two linearly independent sets is lincarly independent (g) The intersection of linearly independent sets is linearly independent (h) Any set containing the zero vector cannot be a basis (i) If there exists a set {v1, ..., v,} that spans V, then dim V < p (j) If there exists a linearly independent set {v1,..., Vp} then dim V > p (k) If dim V = p, then there exists a spanning set of p+1 vectors in V.
Transcribed Image Text:9. Let V be a vector space. Decide if the following statements are true or false. Justify your claims. (a) The empty set is lincarly independent. (b) Any subset of a linearly independent set is linearly independent. (c) Any superset of a linearly dependent set is linearly dependent. (d) The zero set is linearly independent. (e) Any one-element set is linearly independent. (f) The union of two linearly independent sets is lincarly independent (g) The intersection of linearly independent sets is linearly independent (h) Any set containing the zero vector cannot be a basis (i) If there exists a set {v1, ..., v,} that spans V, then dim V < p (j) If there exists a linearly independent set {v1,..., Vp} then dim V > p (k) If dim V = p, then there exists a spanning set of p+1 vectors in V.
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