Use the Basis Test to decide if the following is a basis, Justify your answer {x

Question

Use the Basis Test to decide if the following is a basis, Justify your answer {x2, x2 - x + 1, 2x3 - x2 + 4, x + 11, x2 + x + 9} in P3. {(1,0,2,0), (0,-2,3,1), (2,1,3,-1)} in R4, Find a basis and the dimension of the space spanned by the vectors (1,2,0,1), (0,0,-2,1), (1,0,-1,1). Using the Wronskian determine if {1,e-2x,e3x} is linearly independent. Determine the polynomial, whose graph passes through the points (-3,-1),(-1,2), (1,-1) Find the transition matrix from B to B, if B = {(1,1,1), (1,0,1),(0,0,1)} and B = {(1,1,0), (0,-1,0), (1,0,1)}. Using the transition matrix from 7a), find [x]B, if [x]B = Show that W = {(x,y) R4 : 7x - 5y = 0} is a subspace of R2. Show that W = is NOT a subspace of M22. Let A = Using properties of determinants, find |A-1|, Use Gram-schmidt orthonormalization to transform the basis B = {(1,1,0), (-1,2,1), (0,1,-1)} into an orthonormal basis Find the area of the parallelogram with side vectors u = (1,2,3)

Explanation / Answer

B={(u1,u2,u3)}

u1=(1,1,0)

u2=(-1,2,1)

u3=(0,1,-1)

w1=u1=(1,1,0)

w2=u2-((u2.w1)/||w1||^2)w1

=> w2 =(-1,2 ,1)-(((-1,2,1).(1,1,0))/||1,1,0||^2)(1,1,0)

w2=(-1,2,1) - (1/2)*(1,1,0)

w2=(-3/2,3/2,1)

w3=u3-((u3.w1)/||w1||^2)w1- ((u3.w2)/||w2||^2)w2

w3=(0,1,-1) - (((0,1,-1).(1,1,0))/||1,1,0||^2)(1,1,0) - (((0,1,-1)(-3/2,3/2,1))/||-3/2,3/2,1||^2)(-3/2,3/2,1)

w3=(0,1,-1) - 1/2(1,1,0) - 1/11 (-3/2,3/2,1)

w3 = (-4/11,4/11,-12/11)

v1 =(1/||w1||)w1 = (1/sqrt(2))(1,1,0)

v2=(1/||w2||)w2 = (1/sqrt(11/2))(-3/2,3/2,1)

v3 = (1/||w3||)w3 =(1/sqrt(176))(-4,4,-12)

the orthonormal Basis is (v1,v2,v3).

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