# Row vectors of B and the subspace spanned by row vectors of B lie in row space A:Why?

• Jun 18th 2011, 09:30 AM
x3bnm
Row vectors of B and the subspace spanned by row vectors of B lie in row space A:Why?
I've some questions about the following theorem. The theorem is:

If an m x n matrix \$\displaystyle A\$ is row equivalent to an m x n matrix \$\displaystyle B\$, then the row space of \$\displaystyle A\$ is equal to the row space of \$\displaystyle B\$.

The proof is like this:

Because the rows of \$\displaystyle B\$ can be obtained from the rows of \$\displaystyle A\$ by elementary row operations (scalar multiplication and addition), it follows that the row vectors of \$\displaystyle B\$ can be written as linear combinations of the row vectors of \$\displaystyle A\$.

The row vectors of \$\displaystyle B\$ lie in the row space of \$\displaystyle A\$, and the subspace spanned by the row vectors of \$\displaystyle B\$ is contained in the row space of \$\displaystyle A\$. (<-----My question why's that the row vectors of \$\displaystyle B\$ lie in the row space of \$\displaystyle A\$, and why's the subspace spanned by the row vectors of \$\displaystyle B\$ is contained in the row space of \$\displaystyle A\$?)

But it is also true that the rows of \$\displaystyle A\$ can be obtained from the rows of \$\displaystyle B\$ by elementary row operations. So, you can conclude that the two row spaces are subspaces of each other, making them equal.

-----------End of proof---------

Can anyone kindly answer the marked question in above proof?

If the row vecors of \$\displaystyle B\$ can be written as linear combination of row vectors of \$\displaystyle A\$ how it makes the statement "The row vectors of \$\displaystyle B\$ lie in the row space of \$\displaystyle A\$, and the subspace spanned by the row vectors of \$\displaystyle B\$ is contained in the row space of \$\displaystyle A\$" true?

Is it possible to elaborate on this?
• Jun 18th 2011, 10:15 AM
Deveno
Re: Row vectors of B and the subspace spanned by row vectors of B lie in row space A:
the row space of A IS linear combinations of the rows of A. we got every row-vector of B by adding together scalar multiples of rows of A.

that is precisely how you form linear combinations of vectors.
• Jun 18th 2011, 10:31 AM
x3bnm
Re: Row vectors of B and the subspace spanned by row vectors of B lie in row space A:
Quote:

Originally Posted by Deveno
the row space of A IS linear combinations of the rows of A. we got every row-vector of B by adding together scalar multiples of rows of A.

that is precisely how you form linear combinations of vectors.

Thanks a thousand time Deveno. You showed me the way again. Thank you for that.