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Suppose that G E Fxn. Let X1, X2, ...Xe be a basis for Nul(G). Form a matrix H whose columns are X1, X2, . .. , Xe. 1. Prove that Row(G) C left null space(H). 2. Prove that dim Row(G) =n – l. 3. Prove dim left null space(H) = n – l.

Suppose that G E Fxn. Let X1, X2, ...Xe be a basis for Nul(G). Form a matrix H whose columns are X1, X2, . .. , Xe. 1. Prove that Row(G) C left null space(H). 2. Prove that dim Row(G) =n – l. 3. Prove dim left null space(H) = n – l.

Advanced Engineering Mathematics
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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Suppose that G E Fmxn. Let X1, X2, . .. X, be a basis for Nul(G). Form a matrix H whose columns are X1, X2,... , Xe. 1. Prove that Row(G) C left null space(H). 2. Prove that dim Row(G) = n – l. - 3. Prove dim left null space(H)=n – l. 4. Use the above to prove that Row(G) = left null space(H) The last problem implies that XH = 0 iff X belongs to the Row(G). The above results are true if F, is replaced by R or C.
Transcribed Image Text:Suppose that G E Fmxn. Let X1, X2, . .. X, be a basis for Nul(G). Form a matrix H whose columns are X1, X2,... , Xe. 1. Prove that Row(G) C left null space(H). 2. Prove that dim Row(G) = n – l. - 3. Prove dim left null space(H)=n – l. 4. Use the above to prove that Row(G) = left null space(H) The last problem implies that XH = 0 iff X belongs to the Row(G). The above results are true if F, is replaced by R or C.
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