# Thread: Two questions concerning linear maps/

1. ## 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.

2. ## Re: Two questions concerning linear maps/

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 $T:V\to W$ is surjective if and only if there exists bases $B,B'$ for $V,W$ respectively such that $T:B\to B'$ is a surjection. So, what if you started with a surjection $\{x_1,\cdots,x_n\}\to\{y_1,\cdots,y_m\}$ (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 $v_j$ say as $(a_1,\cdots,a_n)$. Then, you know that $\text{id}(v_j)=(a_1,\cots,a_n)=a_1e_1+\cdots+a_n e_n$ so that I can bet you may conclude that the $j^{\text{th}}$ column of the matrix should just be $v_j$.