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in proving one of these statements. Recall: If PQ, then the contrapositive !Q!P is logically equivalent (the exclamation mark represents the negation of the statement). (a) If 0₁, 02, 03, 0₁ are in R¹ and it is known that {1, 2, 3} is linearly independent then {1, 02, 03, ₁} is linearly independent. (b) If 1, 2, 3, 4 are in R¹ and it is known that {₁, 02, 03,, ₁) is linearly dependent then {1, 2, 3) is linearly dependent.

in proving one of these statements. Recall: If PQ, then the contrapositive !Q!P is logically equivalent (the exclamation mark represents the negation of the statement). (a) If 0₁, 02, 03, 0₁ are in R¹ and it is known that {1, 2, 3} is linearly independent then {1, 02, 03, ₁} is linearly independent. (b) If 1, 2, 3, 4 are in R¹ and it is known that {₁, 02, 03,, ₁) is linearly dependent then {1, 2, 3) is linearly dependent.

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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5. Prove or disprove the following statements. To disprove a statement you only need to provide a counter example. To prove a statement you will need to use the definition and/or theorems. Hint: The contrapositive is very useful in proving one of these statements. Recall: If P→ Q, then the contrapositive !Q →!P is logically equivalent (the exclamation mark represents the negation of the statement). (a) If 0₁, 02, 03, 0₁ are in R¹ and it is known that {₁, 2, 3} is linearly independent then {₁, 2, 3, ₁} is linearly independent. (b) If V1, V2, U3, U₁ are in R¹ and it is known that {V₁, V2, U3,, U₁} is linearly dependent then {1, 2, 3} is linearly dependent.
Transcribed Image Text:5. Prove or disprove the following statements. To disprove a statement you only need to provide a counter example. To prove a statement you will need to use the definition and/or theorems. Hint: The contrapositive is very useful in proving one of these statements. Recall: If P→ Q, then the contrapositive !Q →!P is logically equivalent (the exclamation mark represents the negation of the statement). (a) If 0₁, 02, 03, 0₁ are in R¹ and it is known that {₁, 2, 3} is linearly independent then {₁, 2, 3, ₁} is linearly independent. (b) If V1, V2, U3, U₁ are in R¹ and it is known that {V₁, V2, U3,, U₁} is linearly dependent then {1, 2, 3} is linearly dependent.
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