Let A be an mxn matrix, and let u,ve R", such that Au = 0 and Av = 0. (a) Show that A(u + v) must be the zero vector. Do not skip steps. State theorems used in supporting reasoning. (b) Show that A(cu + dv) = 0 for scalars c and d.
Let A be an mxn matrix, and let u,ve R", such that Au = 0 and Av = 0. (a) Show that A(u + v) must be the zero vector. Do not skip steps. State theorems used in supporting reasoning. (b) Show that A(cu + dv) = 0 for scalars c and d.
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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I need help with Question 9 both a and b please
Transcribed Image Text:5. If vi and we are nonzero vectors in IR5 and v₂ is not a scalar multiple of v₁, then {V₁, V₂ } is linearly
independent.
6. If V₁, V2, V3, V4 € R¹ and v₂ is not a linear combination of V1, V3, V4, then {V1, V2, V3, V4} is linearly independent.
7. If V₁, V2, V3, V4 € R4 and {V₁, V2, V3 } is linearly dependent, then {V1, V2, V3, V4 } is also linearly dependent.
8. If V₁, V2, V3, V4 € R4 and { V₁, V2, V3, V4} is linearly independent, then {V₁, V2, V3 } is also linearly independent.
9. Let A be an mxn matrix, and let u,v e R", such that Au = 0 and Av = 0.
(a) Show that A(u + v) must be the zero vector.
Do not skip steps. State theorems used in supporting reasoning.
(b) Show that A (cu + dv) = 0 for scalars c and d.
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