To determine whether the given vectors form a basis of R⁴, we need to check if they are linearly independent and span R⁴.
To check for linear independence, we can create a matrix using the given vectors as columns:
A = [1 1 2 2; 1 2 5 6; 1 3 6 8; 1 2 4 5]
We can then row-reduce the matrix to see if it reduces to the identity matrix:
Row reduced echelon form of A:
[1 0 1 2; 0 1 2 1; 0 0 0 0; 0 0 0 0]
Since there are two rows of zeros, we can conclude that the given vectors are linearly dependent. Therefore, they do not form a basis of R⁴.
To find the dimension of the subspace spanned by the given vectors, we need to find the maximum number of linearly independent vectors among them.
From the row-reduced echelon form of A, we can see that the first two columns contain the pivots, which means that the first two vectors are linearly independent. Therefore, the dimension of the subspace spanned by the given vectors is 2.
