Bases and Subspaces Worksheet Advanced Linear Algebra, Spring 2013 The vector space for this...

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Bases and Subspaces Worksheet Advanced Linear Algebra, Spring 2013 The vector space for this worksheet is P2; the set of polynomials of degree less that or equal to 2. We are given a basis for this vector space S =  1 + t + t 2 ; 1 + t; 1 + t 2 and a subspace W = span  1 + 2t + 3t 2 ; 2 + t; 1 + t + t 2 : Additionally we are given v 2 P2; but in terms of its coordinates: [v]S = XXXXXXXXXX : 1. Does v 2 W? 2. Find a basis for W 3. Extend the previous basis to a basis of P2; and call this basis S 0 : 4. Characterize w 2 W in terms of [w]S0 5. If we know [w]S ; how do we Önd [w]S0?
	
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Advanced Linear Algebra, Spring 2013
The vector space for this worksheet is P ; the set of polynomials of degree
2
less that or equal to 2. We are given a basis for this vector space
 	
2 2
S = 1+t+t ;1+t;1+t
and a subspace
 	
2 2
W =span 1+2t+3t ;2+t;1+t+t :
2 3
1
4 5
Additionally we are given v2P ; but in terms of its coordinates: [v] = 0 :
2
S
1
1. Does v2W?
2. Find a basis for W
0
3. Extend the previous basis to a basis ofP ; and call this basis S :
2
4. Characterize w2W in terms of [w]
0
S
5. If we know [w] ; how do we ?nd [w] ?
0
S S
1

Robert · Accepted Answer

1. Suppose v ∈ W then we can written as linear combination of spanning
element of W .
We have given [v]S =
 10
1
 This gives v = 1.(1 + t + t2) + 0.(1 + t) +
1(1 + t2)
v = 2 + t + 2t2
Now suppose v ∈W , there must exists a, b, c scalars, such that
2 + t + 2t2 = a(1 + 2t + 3t2) + b(2 + t) + c(1 + t + t2)
2 + t + 2t2 = (a + 2b + c) + (2a + b + c)t + (3a + c)t2
Hence we have
3a + c = 2 2a + b + c = 1 a + 2b + c = 2
We have c = 2− 3a, putting this in remaining equation we get
2a + b + 2− 3a = 1 a + 2b + 2− 3a = 2
that is we have
−a + b = −1 2b− 2a = 0
Which is absurd and do not have solution.

Bases and Subspaces Worksheet Advanced Linear Algebra, Spring 2013 The vector space for this worksheet is P2; the set of polynomials of degree less that or equal to 2. We are given a basis for this...

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