{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# HW4 - AMS210.01 Homework 4 Andrei Antonenko Due at the...

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

AMS210.01. Homework 4 Andrei Antonenko Due at the beginning of the class, March 28, 2003 1. Let f : R 3 R 2 such that f ( x, y, z ) = ( x + 2 z, 2 x - y - z ) and g : R 2 R 3 such that g ( x, y ) = (2 x + y, x + 2 y, x + y ) . Find ( f g ) and ( g f ). 2. Check which of the following functions are linear. Justify your answer. (a) f ( x, y ) = ( x, y 2 ) (b) f ( x, y ) = ( | x + y | , x - y ) (c) f ( x, y, z ) = ( x + y, z, 0) (d) f ( x, y ) = (( x + 1) 2 , y + 3) (e) f ( x, y, z ) = ( x + y + z, y + z, z ) 3. Let f : M n,n R be a function which maps any n × n matrix to a sum of its diagonal elements. Determine is this function linear or not. 4. Let V be a vector space of all n × n -matrices, and M be a fixed matrix in V . Which of the following functions T i : V V are linear: (a) T 1 ( X ) = XM (b) T 2 ( X ) = X + M (c) T 3 ( X ) = XM - MX (d) If M is invertible, T 4 ( X ) = MXM - 1 5. Let f : R 3 R 2 such that f ( x, y, z ) = (2 x + y, x + y + z ). (a) Check that f is a linear function. (b) Find the dimension of the kernel of f and its basis. (c) Find the dimension of the image of f and its basis. 1

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

View Full Document
(d) Find the matrix of this function in standard basis.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

HW4 - AMS210.01 Homework 4 Andrei Antonenko Due at the...

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

View Full Document
Ask a homework question - tutors are online