Answer: No, the given transformation T is NOT a linear transformation.
Step-by-step explanation: We are given to determine whether the following transformation T : R² --> R² is a linear transformation or not :
![T(x,y)=(x,y^2).](https://tex.z-dn.net/?f=T%28x%2Cy%29%3D%28x%2Cy%5E2%29.)
We know that
a transformation T from a vector space U to vector space V is a linear transformation if for
∈U and a, b ∈ R
![T(aX_1+bX_2)=aT(X_1)+bT(X_2).](https://tex.z-dn.net/?f=T%28aX_1%2BbX_2%29%3DaT%28X_1%29%2BbT%28X_2%29.)
So, for (x, y), (x', y') ∈ R², and a, b ∈ R, we have
![T(a(x,y)+b(x',y'))\\\\=T(ax+bx',ay+by')\\\\=(ax+bx',(ay+by')^2)\\\\=(ax+bx',a^2y^2+2abyy'+y'^2)](https://tex.z-dn.net/?f=T%28a%28x%2Cy%29%2Bb%28x%27%2Cy%27%29%29%5C%5C%5C%5C%3DT%28ax%2Bbx%27%2Cay%2Bby%27%29%5C%5C%5C%5C%3D%28ax%2Bbx%27%2C%28ay%2Bby%27%29%5E2%29%5C%5C%5C%5C%3D%28ax%2Bbx%27%2Ca%5E2y%5E2%2B2abyy%27%2By%27%5E2%29)
and
![aT(x,y)+bT(x',y')\\\\=a(x,y)+b(x', y'^2)\\\\=(ax+bx',ay+by')\\\\\neq (ax+bx',a^2y^2+2abyy'+y'^2).](https://tex.z-dn.net/?f=aT%28x%2Cy%29%2BbT%28x%27%2Cy%27%29%5C%5C%5C%5C%3Da%28x%2Cy%29%2Bb%28x%27%2C%20y%27%5E2%29%5C%5C%5C%5C%3D%28ax%2Bbx%27%2Cay%2Bby%27%29%5C%5C%5C%5C%5Cneq%20%28ax%2Bbx%27%2Ca%5E2y%5E2%2B2abyy%27%2By%27%5E2%29.)
Therefore, we get
![T(a(x,y)+b(x',y'))\neq aT(x,y)+bT(x',y').](https://tex.z-dn.net/?f=T%28a%28x%2Cy%29%2Bb%28x%27%2Cy%27%29%29%5Cneq%20aT%28x%2Cy%29%2BbT%28x%27%2Cy%27%29.)
Thus, the given transformation T is NOT a linear transformation.
Answer:
first one has 1 solution, second one have infinity solutions, third one has no solutions and the last one has 1 solution
Step-by-step explanation:
Step-by-step explanation:
Accumulated value = $20,000.
A = P + I, P = principal, I = Interest.
I = 200*5*t/100
I = 1000t/100 = 10t
2000 = P + I = 200 + 10t
2000 - 200 = 10t
1800 = 10t, t = 180.
Therefore, it would take her 180 years to save $20,000.