Let T V W be a linear transformation from a vector space V

Let T : V---> W be a linear transformation from a vector space V into a vector space W. Prove that the range of T is a subspace of W

167 (8 points) Let T : V space W. Prove that the range of T is a subspace of W W b e a linear transformation from a vector space Vinto a vector 7, this ere. nce

Solution

We need to show that the range of T meets the three requirements
to be a subspace
• 0W Range{T}
We know that for any linear transformation T(0V ) = 0W , so 0W is in the
range of T .
• Given any two vectors T(u), T(v) Range{T}, T(u) + T(v)Range{T}
Since T is a linear transformation
T(u + v) = T(u) + T(v)
So T(u) + T(v) is also in the range of T .
• Given any vector T(u) Range{T} and c R, cT(u) Range{T}
Again, exploiting the fact that T is linear
T(cu) = cT(u)
So cT(u) is in the range of T
Since Range{T} meets all properties, Range{T} is a subspace of W .