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2) Let V be a vector space over the complex numbers C. In particular, one has...

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2 Let Vbe a vector space over the complex numbers C. In particular, one has a multiplica- tion of elements of V by real numbers R. It is straightforward to verify that the addition in V together with this multiplication by real numbers turns Vinto a vector space over R. We denote this vector space by Va. a) Find dimcCand dim C. b) Suppose that xi,x2, xn is a basis of V. Show that xi, ixi,x2,ixi, ...,xn, ixn is a basis of V c) Show that 2dimcV dim VR. 

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2) Let V be a vector space over the complex numbers C. In particular, one has a multiplica- tion of elements of V by real numbers R. It is straightforward to verify that the addition in V together with this multiplication by real numbers turns V into a vector space over R. We denote this vector space by V3. a) Find dim C and dim, C. b) Suppose that x,, X2,..., Xạ is a basis of V. Show that x1, ix, X2, ix1, ...,Xn, ix, is a basis of VR. c) Show that 2 dimc V = dimg Va. 2) Let V be a vector space over the complex numbers C. In particular, one has a multiplica- tion of elements of V by real numbers R. It is straightforward to verify that the addition in V together with this multiplication by real numbers turns V into a vector space over R. We denote this vector space by V3. a) Find dim C and dim, C. b) Suppose that x,, X2,..., Xạ is a basis of V. Show that x1, ix, X2, ix1, ...,Xn, ix, is a basis of VR. c) Show that 2 dimc V = dimg Va.

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Linear Algebra A Modern Introduction

Linear Algebra A Modern Introduction

ISBN: 978-1285463247

4th edition

Authors: David Poole

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