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Linear Algebra Worksheet 2 Math 108A Fall 2009, TA Grace Kennedy

Summary: Linear Algebra Worksheet 2
Math 108A Fall 2009, TA Grace Kennedy
Course Website: http://math.ucsb.edu/kgracekennedy/F09108A.html
Supplemental Reading: Axler pages 21-34
Span, Linear Independence, Basis, and Dimension
Here, F represents a field. And V is a vector space of finite dimension n.
1. Prove of disprove the following statements.
(a) If {v1, . . . , vm+1} V is linearly dependent, then so is {v1, . . . , vm-1}.
(b) Any subset of linearly independent vectors is linearly independent.
(c) Any collection of n + 1 vectors {v1, . . . , vn+1} V is linearly depen-
(d) A finite dimensional vector space has a finite number of elements.
(e) If W is another n-dimensional vector space, then there exists a bijec-
tion between W and V .1
2. (From Axler page 35) Let U be the subspace of R2
defined by
U = {(x1, x2, x3, x4, x5) R5
| x1 = 3x2 and x3 = 7x4}
Find a basis of U. What is its dimension?


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics