Homework I - Ch 1 Dept of ECE Faculty of Engineering...

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Dept. of ECE, Faculty of Engineering University of Tehran Linear Algebra Homework # I Chapter 1: Vector Spaces Exercises to be handed: marked by *. Due date: Sunday Esfand 9 th 1383. 1. Let V = { (a 1 , a 2 ) : a 1 , a 2 R }. Define addition of elements of V coordinate-wise, and for (a 1 , a 2 ) in V and c R, define: = = 0 ) , ( 0 ) 0 , 0 ( ) , ( 2 1 2 1 c if c a ca c if a a c . Is V a vector space under these operations? Justify your answer. 2. Let V be the set of sequences {a n } of real numbers. For any {a n } and {b n } V and any real number α , define {a n } + {b n } = {a n + b n } and α .{a n } = { α .a n }. Prove that V is a vector space under these operations. 3. Let V and W be vector spaces over a field F. Let Z = { (v , w) : v V and w W }. Prove that Z is a vector space over F under the operations (v 1 , w 1 ) + (v 2 , w 2 ) = (v 1 + v 2 , w 1 + w 2 ) and c.(v , w) = (c.v , c.w) . 4. How many elements are there in the vector space M mxn (Z 2 ) ? 5. Prove that the upper triangular matrices form a subspace of M mxn (F). Notice: An mxn matrix A is called upper triangular if a ij = 0 for i > j. 6. Let W 1 and W 2 be subspaces of a vector space V. Prove that W 1 U W 2 is a subspace of V if and only if W 1 W 2 or W 2 W 1 . * 7.
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This note was uploaded on 02/01/2010 for the course MATH math115a taught by Professor Shalom during the Spring '10 term at UCLA.

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Homework I - Ch 1 Dept of ECE Faculty of Engineering...

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