Let V be a subspace of R^4 and S = {u1,u2,u3} be a basis for V. Suppose v1, v2, v3 are vectors in V such that (v1)S = (1,−2,0), (v2)S =(2,−7,4), and (v3)S =(−3,8,−1).
Show that {v1, v2, v3} is also a basis for V.
1
Expert's answer
2020-03-13T12:20:14-0400
{u1,u2,u3} is basis of V.
c1u1+c2u2+c3u3=0
v1=u1−2u2
v2=2u1−7u2+4u3
v3=−3u1+8u2−u3
We can solve these three equations to get v1,v2,v3 in terms of u1,u2,u3.
Learn more about our help with Assignments: Linear Algebra
Comments
Assignment Expert
16.03.20, 00:56
There are different methods to solve a linear system of equations with
respect to variables u1, u2, u3. For example, Gaussian elimination
(reduction), elimination of variables, Cramer's rule, the inverse
matrix method. Some hints about these methods can be found at
https://web.mit.edu/16.unified/www/FALL/signalssystems/linear_algebra.pdf,
https://www.mathcentre.ac.uk/resources/Engineering%20maths%20first%20aid%20kit/latexsource%20and%20diagrams/5_6.pdf
.
-
15.03.20, 14:45
How do u get the u1=(23v1+2v2+8v3)/3 u2=(10v1+v2+4v3)/3
u3=(11v1+2v2+5v3)/3 Don't quite really understand this part. Could you
kindly elaborate on it?
Comments
There are different methods to solve a linear system of equations with respect to variables u1, u2, u3. For example, Gaussian elimination (reduction), elimination of variables, Cramer's rule, the inverse matrix method. Some hints about these methods can be found at https://web.mit.edu/16.unified/www/FALL/signalssystems/linear_algebra.pdf, https://www.mathcentre.ac.uk/resources/Engineering%20maths%20first%20aid%20kit/latexsource%20and%20diagrams/5_6.pdf .
How do u get the u1=(23v1+2v2+8v3)/3 u2=(10v1+v2+4v3)/3 u3=(11v1+2v2+5v3)/3 Don't quite really understand this part. Could you kindly elaborate on it?