Project #122759 - HW Linear Algebra

1) Let S = {(-2,0,1),(5,-2,1),(11,7,-5),(-1,4,-2),(2,-3,1),}. Show that Span(S)=R^3, but S is not a basis for R^3. Find a subset of S that is a basis for R^3 and prove that your subset is actually a basis for R^3.

 

2) Let F(−∞,∞) represent the set of all real valued functions that are defined on (−∞,∞). Use the subspace theorem to show that the set of all differentiable functions on (−∞,∞) that satisfy f'(x) + 2f(x) = 0 is a subspace of F(−∞,∞).

 

3) Use the subspace theorem to show that {(0,t,s):t,s ∈ R} is a subspace of R^3 and graph the space in R^3. Also, state the dimension of the subspace.

 

4) suppose that {v1, v2, v3} is linearly independent subset of a vector space V with dim(V) = 4 and that v4 is not in span([v1, v2, v3}). Prove that {v1, v2, v3, v4} is a basis for V.

 

 

5) Show that 𝑥!, 𝑥! − 𝑥!, 𝑥! − 𝑥! + 𝑥!, 𝑥! − 𝑥! + 𝑥! − 𝑥, 𝑥! − 1 is a basis for 𝑃4.

 

 

6) Let A=[5 -3 1 -1 , 4 -2 2 2 , 1 2 8 18] which equal a matrix. Then find a basis for the null space of Na and dim(Na).

 

 

7) Let A1 = [1 0, 1 0], A2=[1 1, 0 0], A3=[1 0, 0 1], A4=[0 0, 1 0]

a) Show that {A1, A2, A3, A4} is a basis for M22.

 

b) find the coordinates of A = [6 2, 5 3] with respect to the basis in part (a)

 

8) Suppose that V is a vector space and dim(V) = 4 and W is a subspace of V. Prove directly without simple quoting a theorem that W must have finite dimension.


9) Let A= [1 2 1 0 -1 , 0 5 2 -1 -5 , 2 9 5 -1 -9] which is a matrix. Then find a basis for null space of A.

 

 

10)

Part 1: Show that if 𝑉! 𝑎𝑛𝑑 𝑉! 𝑎𝑟𝑒 𝑠𝑢𝑏𝑠𝑝𝑎𝑐𝑒𝑠 𝑜𝑓 𝑎 𝑣𝑒𝑐𝑡𝑜𝑟 𝑉, 𝑡ℎ𝑒𝑛 𝑉! ∩ 𝑉! 𝑖𝑠 𝑎 𝑠𝑢𝑏𝑠𝑝𝑎𝑐𝑒 𝑜𝑓 𝑉.

 

Part 2: Suppose that V1 is the subspace of R^3 given by V1={(2t-s, 3t, t+2s):t,s ∈ R} and V2 is the subspace of R^3 given by V2 = {(s,t,t):t,s ∈ R}. Then find a basis for V1 ∩ V2 and dim(V1 ∩ V2).

note: for this problem, you can assume that V1, V2 are subspace of R^3 and dont need to prove this fact.


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Subject Mathematics
Due By (Pacific Time) 04/20/2016 11:00 am
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