Math121sol1

# Math121sol1 - Math 121 Linear Algebra and Applications Prof...

This preview shows pages 1–3. Sign up to view the full content.

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.

Unformatted text preview: Math 121: Linear Algebra and Applications Prof. Lydia Bieri Solution Set 1 Posted: Fri. Oct. 12, 2007 Written by: Luca Candelori Exercise 1. (a) TRUE. This is one of the vector space axioms (VS 3). (b) FALSE. We proved this in class but here is a short proof. Suppose there is another 0 vector, call it 0 . Then for any vector x , x + 0 = x x + 0 = x and therefore x + 0 = x + 0 which, by the Cancellation Law, implies that 0 = 0 . (c) FALSE. If x is the zero vector, this certainly does not hold. (d) FALSE. Again, it does not hold if a = 0. (e) TRUE. An element in F n consists of n coordinates t 1 ,...,t n taken from F . For example 1 √ 2 π can be regarded as an element of R 3 . But this is also a matrix with 3 rows and 1 column so, in general, an element of F n can be regarded as a matrix with n rows and 1 column. This is called the columnn vector representation. You can also represent it as a matrix with n columns and 1 row, in which case we call it a row vector, but this is less common. In fact, we will see later that row vectors are used to represent elements of a the dual space, a vector space that stems quite naturally from any other vector space. (f) FALSE. An m × n matrix has m rows and n columns. Don’t get confused! (g) FALSE. If p has degree d and q has degee e , then p + q is still a polynomial, so it is a member of P ( F ). (h) FALSE. If p has degree d and q has degee e , then p + q has degree ≤ max( d,e ), without equality. For example, in P ( R ),- x 2 +3 and x 2- x + √ 7 are both degree 2 polynomials, but when added they give a polynomial of degree 1. (i) TRUE. Write p ( x ) = a n x n + ... + a , where a n is nonzero so that p has degree n . Then c · p ( x ) = ( ca n ) x n + ... + ca is also of degree n , since c,a n 6 = 0 implies that ca n 6 = 0 1 (j) TRUE. A polynomial of degree 0 in P ( F ) has the form a , for a ∈ F . Therefore one element of F determines a polynomial of degree 0 in P ( F ) completely....
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

Math121sol1 - Math 121 Linear Algebra and Applications Prof...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online