# HW2 - C[0 1 deﬁne f g by f g x = f x g x(we know that...

This preview shows page 1. Sign up to view the full content.

Math 601 HOMEWORK 2 1. Show that the set of 2 × 2 matrices with real entries, with usual addition and multiplication with scalars, is a real vector space. Find its dimension and a basis. 2. For each n denote by P n the vector space of polynomials of degree at most n , with coefficients in R : P n = { p ( t ) = a 0 + a 1 t + a 2 t 2 + . . . + a n t n | a j R } . Consider the following linear transformations: S : P 3 → P 4 defined by S ( p ) = p 0 (0) T : P 3 → P 4 defined by T ( p ) = ( x + 2) p ( x ) H : P 4 → P 3 defined by H ( p ) = p 0 ( x ) + p 0 (0) a) Give the formula for S + T , and for 2 T . b) Give the formula for H T . What is its domain? c) Find the matrix representation of H in the basis { 1 , x, x 2 , x 3 , x 4 } of P 4 and { 1 , x, x 2 , x 3 } of P 3 . d) Prove that T is one to one, but not onto. e) Prove that H is onto, but not one to one. 3. Consider the set of continuous functions on the interval [0 , 1]: C [0 , 1] = { f : [0 , 1] R | f continuous } with the natural addition: for
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: C [0 , 1] deﬁne f + g by ( f + g )( x ) = f ( x )+ g ( x ) (we know that then f + g is also continuous) and natural multiplication by scalars: if c ∈ R and f ∈ C [0 , 1] then cf ∈ C [0 , 1] is deﬁned as ( cf )( x ) = cf ( x ). a) Show that C [0 , 1] is a vector space over R . What is its dimension? b) (BONUS extra 10p.) Show that the functions e x , e 2 x , e 3 x are linearly independent in C [0 , 1]. c) Assuming b) is true, let V = Sp ( e x , e 2 x , e 3 x ) subspace of C [0 , 1], and let T : V → V be given by T ( p ) = p ( x ). Show that T is invertible and ﬁnd T-1 . ( Hint: ﬁnd T-1 on a basis, then extend by linearity to its domain V .) 1...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern