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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.

Linear Algebra: A Modern Introduction
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 21EQ
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I need help with Question 9 both a and b please 

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.
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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