Why row vectors equal to linear combination of basis vectors of rowspc(in this case)?

Let be the row vectors of a matrix:

Suppose the row space of has dimension and basis where .

Using this basis, you can write the row vectors of as(the book wrote this)

My question: why and and ?

Why the book states that the linear combination of basis vectors equal to ?

What method they used to find this? I can't see it. Sorry.

Re: Why row vectors equal to linear combination of basis vectors of rowspc(in this ca

We know that a set of vectors in a vector space is called a basis for if the following conditions are true.

1) spans

2) is linearly independent

I can't find the answer of my above mentioned question. All I can think of is this.

So in the above post are the row vectors of matrix and so spans the row space of which is :

Shouldn't because is the basis of row space of and for the fact that ?

So how and why

?

Can anyone kindly help me understand please?

Re: Why row vectors equal to linear combination of basis vectors of rowspc(in this ca

Re: Why row vectors equal to linear combination of basis vectors of rowspc(in this ca

Quote:

Originally Posted by

**Joanna** are a basis for the rowspace of A, so every combination of the rows of A can be written in terms of this basis.

Each of

is in the row space of A, and so can be written in terms of

.

Did you understand column spaces? Row space works the same way, except obviously related to rows rather than columns. If you did understand column space,

note that the row space of

is equal to the column space of

, and a basis for the row space of

is a basis for the column space of

Thanks a lot for clarifying this.