for what values of a,b,c, and d does the set ( ax+b, cx+d) for a vector space If

Question

for what values of a,b,c, and d does the set ( ax+b, cx+d) for a vector space

If all of them are zero, the set only consists of the zero vector and that is a vector space
Do we have any other case?
My ta said if x= -b/a = -d/c. Then the set has a zero vector
I do agree with that. However I still believe the set is not closed under addition
Pick x1 and x2 and say a,b,c,d are all 2. Then
V1= (2x1 + 2, 2x2 +2). V2 = (2x2 + 2, 2x2+2).
V1+v2= (2(x1+x2) + 4, 2(x1+x2)+4). Therefore it is not closed under addition. Whcih is why I thought they all have to be zero
Am I wrong or is my ta wrong?

Explanation / Answer

Accutally the question means for what values of a,b,c, and d does the set (ax+b,cx+d) for a vector space. for any values of a,b,c and d we have a only a single vector. it can be form a vectors space. if it forms a vector space there will be x=-b/a and y=-d/c. Otherwise (x not = -b/a and y not = -d/c) If you take x1,x2 nd corresponding two vectors V1, V2. as per our disscusion V1=V2. and V1+V2=2(V1) is not equal to V1 surly . then the set is not closed under vector addition. Also we can say that for any scalar exepct 1 we get fails the closed property under scalar multiplication.  

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