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 ₁, U2, U3, U4 are in R4 and it is known that {₁, U2, U3} is linearly independent then {₁, U2, U3, U4} is linearly independent. (b) If 1, U2, U3, U4 are in R4 and it is known that {1, U2, U3,, U4} is linearly dependent then {1, 72, 73} is linearly dependent.
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 ₁, U2, U3, U4 are in R4 and it is known that {₁, U2, U3} is linearly independent then {₁, U2, U3, U4} is linearly independent. (b) If 1, U2, U3, U4 are in R4 and it is known that {1, U2, U3,, U4} is linearly dependent then {1, 72, 73} is linearly dependent.
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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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 ₁, U2, U3, U4 are in R4 and it is known that {₁, U2, U3} is linearly independent then {₁, U2, U3, U4} is linearly
independent.
(b) If 1, U2, U3, U4 are in R4 and it is known that {1, U2, U3,, U4} is linearly dependent then {1, 72, 73} is linearly
dependent.
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