Consider the subset V = {[[a],[b],[c]] | c=(a+b)/2}Show that V is a subspace of R^3 by describing V geometrically.

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mathsworkmusic's profile pic

mathsworkmusic | (Level 2) Educator

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`V = {(a,b,c) | c = (a+b)/2}` .

Now, if `{(x,y,z)} equiv RR^3`

then `V` is a subspace of `RR^3` (3-dimensional space) with the constraint that `z = (x+y)/2`.

This describes a flat plane with symmetry in the line `y=x` and the plane `z=0`. For every unit increase in `x+y`, `z` increases by 1/2.

V is a plane in R3 (3D space)

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jeew-m | College Teacher | (Level 1) Educator Emeritus

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Sub space is a set of vectors. But there are two conditions that shouldbe satisfied inorder to be a subspace.

That is the set of vectors to be;

1.closed under addition

2.closed under multiplication

3.Geometrically a sub space always passes through origin

 

V = {[[a],[b],[c]] | c=(a+b)/2}

Geometrically a sub space always passes through origin.

Now let us consider two vectors a and b.

Assume these two are at the adjacent side of a parallogram. So (a+b) will represent the diagonal.

Now vector c = (a+b)/2 reveals that c is parallel to (a+b).

When a = 0 and b = 0 both at origin then c = (0+0)/2 = 0.

This means c is also passes through origin.

 

Since it is given that c = (a+b)/2 addition in V is already satisfied.

 

When you consider any scaler like x then (ax+bx)/2 = cx.

So scaler multiplication also satisfied.

All three condition to be a sub space is satisfied by V.

 

So V is a sub space of R^3

 

Sources:
rakesh05's profile pic

rakesh05 | High School Teacher | (Level 1) Assistant Educator

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Given V={[[a],[b],[c]] :c=(a+b)/2}.

If consider vector a and vector b to be the adjacent sides of a parallelogram then a+b will represent the diagonal of the parallelogram passing through the joining of the vectors Vector a  and  vector b. and (a+b)/2 is a vector parallel to the vector (a+b).

Clearly 0 is an element of V as the vector c will pass through (0,0)

as (0+0)/2=0

Also addition of two vectors is already permitted in V. Now if we take any scalar i.e. any real number say x we see that

(xa+xb)/2=xc.

So all the conditions of being a vector space are satisfied by V. Hence V is a subspace of R^3.

 

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