Consider the following two bases for R3 : α:={[213],[−101],[31−1]}and β:={[111],[−231],[23−1]} If [x]α=[12−1]αthen find [x]β (that is, express x in the β coordinates).
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α={[213],[−101],[31−1]}β={[111],[−231],[23−1]} [x]α=[12−1]α→x=1.[213]+2[−101]−1[311]=[−306] Let x=C1[111]+C2[−231]+C3[23−1]⇒C1−2C2+2C3=−3−(1)C1+3C+2+3C3=0−(2)C1+C2−C3=6−(3) From (3) C3=C1+C2−6 From (1) C1−2C2+2C1+2C2−12=−3⇒3C1=+9⇒C1=3 From (2) C1+3C2+3C1+3C2−18=0⇒4C1+6C2=18⇒6C2=C2 C3=3+1−6=−2 So [−306]=+3[111]+1[−231]−2[23−1] So [x]β=[31−2]
(a) Find the bases and dimension for the subspace H={[3a+6b−c6a−2b−2c−9a+5b+3c−3a+b+c];a,b,c∈R} (b) Let be bases for a vector space V,and suppose (i) Find the change of coordinate matrix from B toD. (ii) Find [x]Dforx=3b1−2b2+b3
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The system of equation {2x+y=14x+2y=3 by graphing method and if the system has no solution then the solution is inconsistent. Given: The linear equations is {2x+y=14x+2y=3
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