a) Is it true that the span of a non-zero vector in R^2 is a line through the or
ID: 3036485 • Letter: A
Question
a) Is it true that the span of a non-zero vector in R^2 is a line through the origin? b) Is it true that the span of any two non-zero vectors in R^2 is R^2? If not, write two non-zero vectors that do not span R^2. c) Is it true that the span of any two non-zero vectors in R^3 is a plane though the origin? If not, write two non-zero vectors of R^3 that do not span a plane though the origin. d) Can you find 4 linearly independent vectors is R^3? If so, write 4 linearly independent vectors in R^3. e) Can you find 4 vectors of R^3 that span R^3? If so, write 4 vectors in R^3 that span R^3.Explanation / Answer
(a) In R2, the span of any non-zero vector is the line through that vector. It may or may not pass through the origin. The statement is False.
(b) The statement is False. The span of any 2 non-zero vectors in R2 is R2 only when the 2 vectors are linearly independent. An example is ( 1,1)T and ( 2,2)T.
(c) The statement is False. If two vectors are linearly dependent their span is the line determined by the vector. An example is ( 1,1,0)T and (2,2,0)T.
(d) The statement is False. The maximum number of linearly independent vectors in R3 is 3.
(e) The statement is True. An examople is (1,0,0)T, (0,1,0)T, (0,0,1)T, (1,2,3)T.
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