Exam1_Math121

# Exam1_Math121 - Midterm 1 for Math 121 Fall 2006 Wednesday...

This preview shows pages 1–2. Sign up to view the full content.

Midterm 1 for Math 121, Fall 2006. Wednesday October 25 Time allowed: 53 minutes Apart from the ﬁrst problem you must fully justify your answers. You may assume all vector spaces are ﬁnite-dimensional unless otherwise stated. Recall that for a linear operator T : V V , the notation N ( T ) denotes the null space and R ( T ) denotes the range. 1. (20 points) Mark the following statements true or false. No justiﬁcation is needed. (a) Let W 1 ,W 2 be two vector subspaces of V and β 1 2 be bases of W 1 ,W 2 respec- tively. Then β 1 β 2 is a basis of W 1 W 2 . (b) Let P ( R ) denote the (inﬁnite-dimensional) vector space consisting of all poly- nomials with real coeﬃcients. Let g ( x ) ∈ P ( R ) be some ﬁxed polynomial. Then the function T : P ( R ) → P ( R ) given by T ( p ( x )) = g ( x ) p ( x ) is linear. (c) Let V be a vector space and T,U : V V be two linear operators. Then N ( T ) N ( TU ). (d) Let

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 09/09/2009 for the course MATH 350 taught by Professor Hoelscher during the Spring '08 term at Rutgers.

### Page1 / 2

Exam1_Math121 - Midterm 1 for Math 121 Fall 2006 Wednesday...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online