Assertion: matrix's vector | /r/HasProperty | independent
Facets:
value: linearly
type: DEGREE
Source sentences:
text: In particular, the vectors are linearly independent if and only if the parallelotope has nonzero n-dimensional volume, if and only if Gram determinant is nonzero, if and only if the Gram matrix is nonsingular.
text: We claim that the vectors A x 1 , A x 2 , … , A x r {\displaystyle Ax_{1},Ax_{2},\ldots ,Ax_{r}} are linearly independent.
text: Therefore, this matrix has two linearly independent vectors and as a consequence, the other vector is dependent on the other two.
text: We have already triangulated the matrix and we have not left any row with zeros, then the 3 vectors of the matrix are linearly independent.
text: Therefore, as the matrix has 3 linearly independent vectors, its range is 3:
text: If the element a-4 is zero, from where a=4 is obtained, the matrix would have two linearly independent vectors, then its range would be 2:
text: For any other value of a than 4, i.e., that the element a-4 was different from zero, then the matrix has 3 linearly independent vectors and therefore its ranto is 3: