3. Let V be a subspace of R" and B:= {v1,..., V} be a basis for V. Prove that each vector in V can be written uniquely as a linear combination of vectors in B.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.3: Spanning Sets And Linear Independence
Problem 22EQ
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please help with question 3 only

Fürther, if A is invertible (m = n), then row(AB) = row(B).
(b) Show that rank(AB) < rank(B).
Further, if A is invertible (m = n), then rank(AB) = rank(B).
|3D
3. Let V be a subspace of R" and B:= {v1, ..., Vg} be a basis for V.
Prove that each vector in V can be written uniquely as a linear combination of vectors in B.
1 1
-2
1.
4. Let A := 0 1
0 -1
23 -4
1.
(a) Use the algorithm on page 202 to find a basis for column/row/null spaces of A.
(b) Use the reduced row echelon form of A in (a) to write the 4-th column vector of A as a
linear combination of the others.
Transcribed Image Text:Fürther, if A is invertible (m = n), then row(AB) = row(B). (b) Show that rank(AB) < rank(B). Further, if A is invertible (m = n), then rank(AB) = rank(B). |3D 3. Let V be a subspace of R" and B:= {v1, ..., Vg} be a basis for V. Prove that each vector in V can be written uniquely as a linear combination of vectors in B. 1 1 -2 1. 4. Let A := 0 1 0 -1 23 -4 1. (a) Use the algorithm on page 202 to find a basis for column/row/null spaces of A. (b) Use the reduced row echelon form of A in (a) to write the 4-th column vector of A as a linear combination of the others.
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