Find the dimension of the subspace U = span{[1, 1, 1], [2, 5, 2], [1, 2, 3]}.Select one:a. 1b. none of the other choices is truec. 3d. 2

To find the dimension of the subspace U, we need to find the rank of the matrix formed by the given vectors. The vectors are [1, 1, 1], [2, 5, 2], and [1, 2, 3]. We form a matrix with these vectors as rows:

1 1 1 2 5 2 1 2 3

We can simplify this matrix using Gaussian elimination:

Subtract the first row from the second and third rows:

1 1 1 1 4 1 0 1 2

Subtract the second row from the first:

0 -3 0 1 4 1 0 1 2

Subtract 4 times the third row from the second:

0 -3 0 1 0 -7 0 1 2

Now, we can see that the second and third rows are linearly independent, and the first row is a linear combination of the second and third rows. Therefore, the rank of the matrix is 2, and the dimension of the subspace U is 2. So, the correct answer is d. 2.