Solve using the concept of rank.
Is
S={−16 −7 −21,2 1 3, 21 9 2}
a linearly independent set of vectors in R3?
So I know how to find out if this set is linearly indep. But, I don't know how to use row rank to find this out..
To use the concept of rank to determine if a set of vectors is linearly independent, we can form a matrix using the given vectors as its rows (or columns). Then, we can calculate the rank of this matrix.
In this case, the set of vectors is:
S = { (-16, -7, -21), (2, 1, 3), (21, 9, 2) }
To use row rank, we can form a matrix by writing the vectors as rows. We get the matrix:
M =
[ -16 -7 -21 ]
[ 2 1 3 ]
[ 21 9 2 ]
To find the rank of this matrix, we can perform row operations to reduce it to row-echelon form or reduced row-echelon form using Gaussian elimination or other row reduction techniques. The row rank is the number of non-zero rows in the row-echelon form or reduced row-echelon form.
Applying Gaussian elimination,
M =
[ -16 -7 -21 ]
[ 2 1 3 ]
[ 21 9 2 ]
Multiply the second row by 8 and add it to the first row:
M' =
[ 0 -13 -5 ]
[ 2 1 3 ]
[ 21 9 2 ]
Multiply the third row by -7 and add it to the first row:
M'' =
[ 0 -13 -5 ]
[ 2 1 3 ]
[ 0 -48 -39 ]
The rank of this matrix is 2 because there are 2 non-zero rows in the reduced row-echelon form. Therefore, the row rank of the matrix is 2.
Now, since the row rank of the matrix is equal to the number of non-zero rows in the matrix, we can conclude that the vectors in the given set S are linearly independent if and only if the row rank of the matrix is equal to the number of vectors in the set.
In this case, the row rank of the matrix is 2, which is equal to the number of vectors in the set (3). Therefore, we can conclude that the set S={ (-16, -7, -21), (2, 1, 3), (21, 9, 2) } is linearly independent in R^3.