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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:8. 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, ..., Uz}. Form a kxn matrix B with u1, u2, ., Uz
***.
as its rows.
Show that nullspace of B is equal to V+.
(iii) Show that dim V + dim V+ = n.
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