Understanding Basis of Vector Spaces
Explore the concept of basis in a vector space explained by Prof. Uday Raj Singh, including the definition, properties, and an illustrative example with a standard basis for V3(R). Learn how to determine if a set of vectors forms a basis and its significance in generating the vector space.
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Presentation Transcript
Topic Basis of a vector space by ;- prof. (dr.) uday raj singh
Basis of a Vector Space IF V(F) IS A VECTOR SPACE AND S IS ANY SUBSETOF V(F), THEN S IS CALLED ABASIS FOR V(F) IF : S IS L.I. EVERY VECTOR OF V(F) IS EXPRESSIBLE AS THE (i) (ii) LINEAR COMBINATION OF VECTORS OF S UNIQUELY. I.E., S GENERATES V(F) I.E., L(S) = V(F).
Example 1. A set of vectors S = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} is standard basis for V3(R). Sol. (a)Firstly,we checkSforL.I. So let, 1, 2, 3 R such that 1(1, 0, 0) + 2(0, 1, 0) + 3(0, 0, 1) = 0 ( 1, 0, 0) + (0, 2, 0) + (0, 0, 3) = 0 ( 1, 2, 3) = (0, 0, 0) 1= 0 .. . (i) 2= 0 . (ii) 3= 0 (iii)
Let M is coefficients matrix of above equations M = |M|= 1 0 Thus, S is I.I.
(b) Let (x, y, ) V3(R) (x, y, ) = x(1, 0, 0) + y(0, 1, 0) + (0, 0, 1) = A linear combinationof members of S. S generates V3(R) i.e., L(S) = V3(R) Hence, S is basis for V3(R).
