Alg1231W4T3 - Linear Independence of Polynomials

Friday, 17 August 2018

1:03 AM

Machine generated alternative text: i) The set -3m + 1,5x2 -2m + 1,2x2 equation 3 x 2 of polynomials is a linearly — 3m + 1) -F 13 ( 5 p — 1) O subset of IP2 . This is because the independent has only the zero 2 ii) If 2m equation has a non-zero O solution given by [a,ß, 7'] is replaced by —IOC — 9T+ 2, then the set above is now a linearly dependent 0 subset of IP2 . This is because the 2 O solution given by [a,ß, 7'] Note: enter your coefficients above using square brackets (e.g. [1 , 2 2 _ 2 x+l —10m ,3])

 

Machine generated alternative text: A common example of linearly independent polynomials is given by polynomials of different degrees. For example in IP3 the set 3 is a linearly independent O set. Let's see why. Suppose that Cl 1) + % (X + C4(x Here the zero on the right-hand side of the equation means the zero polynomial in IP3, namely 3 —o. Expanding the powers and comparing the coefficients of x on both sides we deduce that we must have c4 Using this fact and comparing the coefficients of x on both sides we further deduce that c3 — 0 Using both of these facts and comparing the coefficients of on both sides, we get c2 Finally, using all three facts above and comparing the remaining constant terms allows us to deduce that Cl — 0 Hence the only solution O is given by Cl = = = c4 3 is a linearly independent set. 0 O , and so we have proven that

 

Machine generated alternative text: In this problem we are working in the vector space IP2 of polynomials of degree two or less i) We can show that {1,@ + 1), (x + 1) , x is a linearly dependent set by writing 2 ii) Let's also show that {1, (x — 1), (x —1) is linearly dependent by writing 2 — 1)2.

Machine generated alternative text: Answer the following 'True or False' questions (note: even without the magic 'how did I do' button, these questions should not require any calculation). i) The set O True @ False is linearly independent. ii) The set is linearly independent. O True @ False iii) The set is linearly dependent. @ True O False

 

Machine generated alternative text: iv) The set {0} is linearly independent. O True @ False v) The set {—9,4 + 9 O True @ False 9 x} is linearly independent vi) The set {0, x} is linearly dependent @ True O False

 

 

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