Thread: Two questions concerning linear maps/

1. Let V be a vector space of dimension n, and W be a vector space of dimension m <= n. Show that there exists a surjective linear map V -> W.

2. Let id denote the identity map R^n -> R^n. Let B denote a basis B = {v1, v2, ..., vn} of R^n and let E denote the standard basis E. Compute the matrix representation of I with respect to B, E.

It is obvious to me that a surjective linear map exists if W is of lower dimension than V. I am unsure of how exactly to show this, though. For the map f: V -> W to be surjective, an f(v) (such that v is an element of V) must exist for every w (w is an element of W).

I'm unsure of what I'm attempting to do on the second problem.

Originally Posted by pantsaregood

1. Let V be a vector space of dimension n, and W be a vector space of dimension m <= n. Show that there exists a surjective linear map V -> W.

So think about this, a linear map is surjective if and only if there exists bases for respectively such that is a surjection. So, what if you started with a surjection (where these are the obvious bases from our problem) and........

2. Let id denote the identity map R^n -> R^n. Let B denote a basis B = {v1, v2, ..., vn} of R^n and let E denote the standard basis E. Compute the matrix representation of I with respect to B, E.

It is obvious to me that a surjective linear map exists if W is of lower dimension than V. I am unsure of how exactly to show this, though. For the map f: V -> W to be surjective, an f(v) (such that v is an element of V) must exist for every w (w is an element of W).

I'm unsure of what I'm attempting to do on the second problem.

Think about it this way, write say as . Then, you know that so that I can bet you may conclude that the column of the matrix should just be .