Suppose that V and W are both finite dimensional. Prove that there exists a surj
ID: 2941136 • Letter: S
Question
Suppose that V and W are both finite dimensional. Prove that there exists a surjective linear map from V onto W if and only if dim W<= dim V.Explanation / Answer
given that V and W are finite dimensional vector spaces. suppose T: V--> W is a linear transformation. suppose T is a surjective. i.e. the basis of W is spanned by the images set of the basis of V. i.e. the number of linearly independent vectors in the image set of the basis of V is more than the dimension of W. while the basis always has greater or equal to the number of vectors than the linearly independent set of the respective images, it follows that the dimension of V is greater than or equal to the dimension of W. converse of this theorem is not necessarily true as it is established in the following example. T: R^3--> R^2 defined by T(x,y,z) = ( x,0) clearly, T is a linear transformation while T(au+bv) = aT(u)+bT(v) further, the dimension of the range of T is 1 that means T cannot span the entire R^2 so, T is not onto however dimension of R^3 greater than dimension of R^2. if you are stuck , please message me. o.k.? thank you.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.