Assignment 2

# Assignment 2 - A T is invertible and ( A T )-1 = ( A-1 ) T...

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

MATH 223, Linear Algebra Winter 2008 Assignment 2, due in class Wednesday, January 23, 2008 C stands for the set of complex numbers, and R for the set of real numbers. 1. Here are some subsets of the complex vector space C 3 . In each case, decide whether the given set S is a subspace of C 3 ; justify your answers. (a) S = { a - b 3 ib (2 + i ) a - 2 b : a,b any complex numbers } . (b) S = { a - b 3 ib + 2 (2 + i ) a - 2 b : a,b any complex numbers } . (c) S = { a - b 3 ib (2 + i ) a - 2 b : a,b any real numbers } . (d) S = { ~v ∈ C 3 : ± 1 2 3 i 2 i 3 i ~v = ± 0 2 - 2 i 4 i 1 - 4 i 0 ~v } . 2. Let R R be the (real) vector space of all functions from the real numbers to the real numbers. For each of the following subsets of R R , decide whether it is a subspace; justify your answers. (a) The set of all even functions. (A real-valued function is called even if it satisﬁes f ( - x ) = f ( x ) for all x .) (b) { f ∈ R R : f (0) = 7 } . (c) { f ∈ R R : f (7) = 0 } . 3. Suppose that A is an invertible matrix over any ﬁeld. Show that

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: A T is invertible and ( A T )-1 = ( A-1 ) T . 4. Let A = i 1-i and B = 1 2-i 2 i 1 + 4 i , both invertible matri-ces with complex entries. Find AB , A-1 , B-1 , ( AB )-1 , A-1 B-1 and B-1 A-1 . Express A and B and each of these matrices as a product of elementary matrices. 5. Suppose that V is any vector space over any eld, and W 1 and W 2 are subspaces of V . (a) Show that the intersection W 1 W 2 is also a subspace of V . (b) Show that the union W 1 W 2 is almost never a subspace of V specically, show that this only occurs if W 1 W 2 or W 2 W 1 . 1 (c) We let the sum W 1 + W 2 be dened as the set { ~w 1 + ~w 2 : ~w 1 W 1 , ~w 2 W 2 } . Show that this set is a subspace of V . 2...
View Full Document

## This note was uploaded on 04/29/2008 for the course PHYS 230 taught by Professor Harris during the Fall '07 term at McGill.

### Page1 / 2

Assignment 2 - A T is invertible and ( A T )-1 = ( A-1 ) T...

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

View Full Document
Ask a homework question - tutors are online