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HW12 (2) - University of Colorado Department of Physics...

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University of Colorado, Department of Physics PHYS3220, Fall 09, HW#12 due Wed, Nov 18, 2PM at start of class 1. Vectors (Total: 20 pts) a) (Griffiths, Problem A.1) Consider the ordinary vectors in 3D ( a x ˆ i + a y ˆ j + a z ˆ k ), with complex components. For each of the following three subsets find out whether or not it constitutes a vector space. If so, what is the dimension of the vector space? If not, why is it not a vector space? (i) The subset of all vectors with a z = 0. (ii) The subset of all vectors whose z -component is 1 (iii) The subset of all vectors whose components are all equal. b) Does the subset of all 2 × 2 matrices form a vector space? Assume the usual rules for matrix addition and multiplication by a scalar, namely: a b c d + e f g h = a + e b + f c + g d + h , α a b c d = αa αb αc αd If it does not form a vector space, why not? If it does form a vector space, state the dimensionality and give an example of a set of basis vectors.
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