Let V be a subspace of R". Recall (from exercise 5.7) that V- is also a subspace of R". (i) Suppose A is an n x n matrix such that Av e V for any v e V. Show that ATw e V+ for any w e V+. (ii) Suppose V = span{u1, u2 . as its rows. Uz}. Form a kxn matrix B with u1, U2, ..., Uz Show that nullspace of B is equal to V+. (iii) Show that dim V + dim V± = n.

Let V be a subspace of R". Recall (from exercise 5.7) that V- is also a subspace of R". (i) Suppose A is an n x n matrix such that Av e V for any v e V. Show that ATw e V+ for any w e V+. (ii) Suppose V = span{u1, u2 . as its rows. Uz}. Form a kxn matrix B with u1, U2, ..., Uz Show that nullspace of B is equal to V+. (iii) Show that dim V + dim V± = n.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.4: Orthogonal Diagonalization Of Symmetric Matrices

Problem 26EQ

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