- #1

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- Homework Statement
- Show that ##U = span \{ (1, 2, 3), (-1, 2, 9)\}## and ##W = \{ (x, y, z) \in \Bbb R^3 | z-3y +3x = 0\}## are equal.

- Relevant Equations
- N/A

Show that ##U = span \{ (1, 2, 3), (-1, 2, 9)\}## and ##W = \{ (x, y, z) \in \Bbb R^3 | z-3y +3x = 0\}## are equal.

I have the following strategy in mind: determine the dimension of subspaces ##U## and ##W## separately and then make use of the fact ##dim U = dim W \iff U=W##. For ##U## I would construct a ##2 \times 3## matrix and use Gaussian elimination on it. My issues are

1) Is the dimension of the row space the one of interest? If that is the case, one gets that ##dim U = 2## because the Gaussian elimination yields no zero rows.

2) For ##W## I do not see how to get the dimension, because I am not given the basis elements of ##W##.Your guidance is appreciated.

Thanks!

I have the following strategy in mind: determine the dimension of subspaces ##U## and ##W## separately and then make use of the fact ##dim U = dim W \iff U=W##. For ##U## I would construct a ##2 \times 3## matrix and use Gaussian elimination on it. My issues are

1) Is the dimension of the row space the one of interest? If that is the case, one gets that ##dim U = 2## because the Gaussian elimination yields no zero rows.

2) For ##W## I do not see how to get the dimension, because I am not given the basis elements of ##W##.Your guidance is appreciated.

Thanks!