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.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.3: Orthonormal Bases:gram-schmidt Process

Problem 17E: Complete Example 2 by verifying that {1,x,x2,x3} is an orthonormal basis for P3 with the inner...

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Question

please solve all parts or atleast 4th part

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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