e problem. t) Let H\'be the set of all points of the form (s. s-11. Determine wh

Question

e problem. t) Let H'be the set of all points of the form (s. s-11. Determine whether His a vector space. If it is not a vector space, determine which of the following properties it fails to satisfy A: Contains zero vector B: Closed under vector addition C Closed under multiplication by scalars A) H is not a vector space; does not contain zero vector B) H is a vector space. C) H is not a vector space; fails to satisfy all three properties D) H is not a vector space; not closed under vector addition

Explanation / Answer

solution:

Let H be the set of all points of the form (s, s-1). Determinewhether H is a vector space. If it is not a vector space, determinewhich of the following properties it fails to satisfy.

LET ANY 2 VECTORS IN H BE U=[A,A-1] AND V=[B,B-1]

LET K BE ANY SCALAR

A: Contains zero vector

LET THE ZERO VECTOR BE [Z,Z-1].

THEN WE SHOULD HAVE

A+Z=A ....THAT IS

[A+Z,A+Z-2]=[A,A-1]

HENCE A=A+Z....Z=0

A+Z-2=A-1....Z=1...CONTRADICTORY

SO NO ZERO ELEMENT EXISTS.

B: Closed under vector addition

A+B=[A,A-1]+[B,B-1]=[A+B,A+B-2]....NOT AN ELEMENT OF H AS IICOORDINATE SHOULD BE A+B-1.

SO NOT CLOSED IN ADDITION.

C: Closed under multiplication by scalars

K[A,A-1] =[KA,KA-K].......NOT AN ELEMENT OF H AS II COORDINATESHOULD BE KA-1.

SO NOT CLOSED IN SCALAR MULTIPLICATION.

As you can see above h is not a vector space .
It failed to satisfy the above 3 properties.
Please note that failure to satisfy even a single property
is enough to rule it out as a vector space

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