DOC PREVIEW
Berkeley MATH 110 - Linear Algebra Midterm Sample Questions

This preview shows page 1 out of 2 pages.

Save
View full document
Premium Document
Do you want full access? Go Premium and unlock all 2 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Mathematics 110 Name Linear Algebra Midterm Sample Questions Write clearly with complete sentences explaining your work You will be graded on clarity style and brevity If you add false statements to a correct argument you will lose points 1 Let V be a vector space over a field F a What is the definition of a linear subspace of V b What is the definition of the span of a list v1 vn in a vector space V Prove that the span of a list in V is the smallest linear subspace of V containing each element of the list c What is the definition of the dimension of a vector space Explain why this definition make sense 2 Let V and W be vector spaces over a field F a What is the definition of a linear transformation from V to W b If v1 v2 vn is a basis for V and w1 w2 wm is an ordered basis for W what is the definition of the matrix representation M T of a linear transformation from V to W with respect to the bases and c Let V be the space of polynomials of degree at most 2 over R and let 1 x x2 an ordered basis for V Let T V V be the transformation sending p to p0 2p where p0 is the derivative of p Find M T 3 If V and W are vector spaces let L V W denote the set of linear transformations from V to W a Explain the definition of the sum S T of two elements S and T of L V W and in particular show why with your definition S T L V W b Let P denote the space of polynomials over the field of real numbers Explain why the map D P P sending f to its derivative is linear Prove that Id D D2 D3 is linearly independent in L P P 4 Let V be a finite dimensional vector space and let S and T be linear transformations from V to itself Prove that if ST S T then ST T S Show that this need not be true if V is not finite dimensional Hint compute S idV T idV 5 Let V and W be vector spaces over F and let V W be the set of pairs v w where v V and w W Then V W can be made into a vector space using the operations of V and W We use this structure from now on If f V W is a function its graph is the subset of V W consisting of those pairs v w such that w f v Show that f is linear if and only if its graph is a linear subspace of V W 2


View Full Document

Berkeley MATH 110 - Linear Algebra Midterm Sample Questions

Download Linear Algebra Midterm Sample Questions
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Linear Algebra Midterm Sample Questions and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Linear Algebra Midterm Sample Questions and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?