Math Is Fun Forum

virtualinsanity

Member

Registered: 2007-03-11

Posts: 38

Row space

There's this problem in my linear algebra book:

Describe the vectors which are in the row space of the following matrix:

(1 3
2 0
-1 1)

I was wondering how this is done so if anyone could help, that would be great!  Thanks!

ryos

Member

Registered: 2005-08-04

Posts: 394

Re: Row space

I'm never sure what math people mean when they say "describe". I think they probably mean that they want you to find out if they are linearly dependent or not.

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El que pega primero pega dos veces.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Row space

When you’ve found a basis for the space, you will have “described” the vectors in the space (since every vector is a linear combination of the basis vectors).

virtualinsanity

Member

Registered: 2007-03-11

Posts: 38

Re: Row space

When you’ve found a basis for the space, you will have “described” the vectors in the space (since every vector is a linear combination of the basis vectors).

Does this mean that I have to put this matrix in row-reduced echelon form?  As in row reduce the following...

(1  3| 0)
(2  0| 0)
(-1 1| 0)

?

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Row space

ie
a(1 3)+b(2 0) represents any vector composed by row vectors of the Matrix.

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X'(y-Xβ)=0

virtualinsanity

Member

Registered: 2007-03-11

Posts: 38

Re: Row space

Oh, okay.  So I put the matrix in row-reduced echelon form and got the matrix:

1 0 | 0
0 1 | 0
0 0 | 0

Does this mean that the set of vectors in the row space of a are just (0,0) since:

a = 0
b = 0

Therefore, 0(1 3) + 0(2 0) = (0 0) if I'm not mistaken?

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Row space

No, row space means any another vector with the same amount of entries can be represented by a linear combination of the given row vectors from the matrix.

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X'(y-Xβ)=0