Which of the following sets spans the vector spaceR3? Select all that apply. A. ⎧⎩⎨⎪⎪⎡⎣⎢211⎤⎦⎥,⎡⎣⎢301⎤⎦⎥,⎡⎣⎢512⎤⎦⎥,⎡⎣⎢602⎤⎦⎥⎫⎭⎬⎪⎪ B. ⎧⎩⎨⎪⎪⎡⎣⎢110⎤⎦⎥,⎡⎣⎢300⎤⎦⎥⎫⎭⎬⎪⎪ C. ⎧⎩⎨⎪⎪⎡⎣⎢201⎤⎦⎥,⎡⎣⎢110⎤⎦⎥,⎡⎣⎢001⎤⎦⎥⎫⎭⎬⎪⎪ D. ⎧⎩⎨⎪⎪⎡⎣⎢100⎤⎦⎥,⎡⎣⎢012⎤⎦⎥,⎡⎣⎢003⎤⎦⎥⎫⎭⎬⎪⎪

To determine which of the given sets spans the vector space R3, we need to check if the vectors in each set can generate all possible vectors in R3 through linear combinations.

Let's analyze each set:

A. { [2 1 1], [3 0 1], [5 1 2], [6 0 2] } B. { [1 1 0], [3 0 0] } C. { [2 0 1], [1 1 0], [0 0 1] } D. { [1 0 0], [0 1 2], [0 0 3] }

To determine if a set spans R3, we need to check if the vectors in the set are linearly independent. If they are linearly independent, then they span R3.

Let's check each set for linear independence:

A. We can see that the vectors in set A are linearly independent since none of them can be written as a linear combination of the others. Therefore, set A spans R3.

B. The vectors in set B are linearly independent since neither vector can be written as a linear combination of the other. Therefore, set B spans R3.

C. The vectors in set C are linearly independent since none of them can be written as a linear combination of the others. Therefore, set C spans R3.

D. The vectors in set D are linearly independent since none of them can be written as a linear combination of the others. Therefore, set D spans R3.

In conclusion, sets A, B, C, and D all span the vector space R3.