Dear Dr. Tsumura,

If we are ask to determine W = Sp{ vector u, vector v} given vector u and v, could I apply

W = column space of Sp

= row space of Sp^Transpose ?

Hope to hear back from you soon. Thank you.

First of all, the column space of $\Span$ does not make sense. We defined the column space **of a matrix**.

So before talking about the column space, you need to define a matrix $A=[\mathbf{u}, \mathbf{v}]$.

Then you can say $W$ is the column space of $A$. (Or the row space of $A^{\trans}$.)

If you want to find a basis of $W$, then you can use the leading 1 method.

Ok. Noted. Thank you, Dr. Tsumura.

Besides that, I would like to ask if we are ask to prove whether for example W is a subspace of R^3 or not, and I know that W will not be satisfied closure under addition. Could I jump to S2 instead of S1 to save time, same goes to prove whether it is a Vector Space?

I don’t understand your question. What do you mean by “I know that W will not be satisfied closure under addition”? If you are trying to prove W is not a subspace, then you just need to give a counterexample of one of S1-S3. But if you are trying to prove W is a subspace, then you must check all S1-S3 conditions.

Thank you, Dr. Tsumura. You have just answered my question.