once solving a problem around linearly independent/dependent, i don"t know what the answer means.

You are watching: How to tell if a matrix is singular or nonsingular

Question: determine whether the arsenal of vectors is linearly elevation in $R^3$:$(1,2,4)^T$ , $(2,1,3)^T$ , $(4,-1,1)^T$

Answer:If $$c_1(1,2,4)^T+c_2(2,1,3)^T+c_3(4,-1,1)^T=(0,0,0)^T$$

Then,$$eginarrayrclc_1+2c_2+4c_3=0\2c_1+c_2-c_3=0\4c_1+3c_2+c_3=0\endarray$$The coefficient matrix of the mechanism is singular and hence the system has nontrivial solutions. Therefore the vectors are linearly dependent.

I know the technique to recognize whether the procession is linearly live independence or no by computing $c_1v_1+c_2v_2+...c_nv_n=0$, yet here ns don"t recognize why the mechanism is singular, for this reason how have the right to I tell even if it is it"s singular or not? are there any type of ways to know?


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edited jan 3 "19 at 3:59
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EuxhenH
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asked jan 3 "19 in ~ 3:46
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If the determinant the the coefficient procession is zero, then the procession is singular and the mechanism in dependent. The homogeneous mechanism in this case has a non-zero solution and also the trivial zero solution.

Otherwise the matrix is non-singular and also the system has a distinct solution i beg your pardon in case of homogeneous device is $(0,0,0)^T$


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answered jan 3 "19 at 3:57
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Mohammad Riazi-KermaniMohammad Riazi-Kermani
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There are numerous approaches that have the right to be take (some have already been provided). One other is to apply Gaussian elimination so that the procession is in row echelon form (commonly denoted together $operatornameREF(A)$).

If the $operatornamerankleft(operatornameREF(A) ight) eq n$ climate $A$ is not invertible and the device of equations is singular.


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edited january 3 "19 in ~ 7:16
answered january 3 "19 at 4:06
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