Question

determine whether the vector v = (4, 3, 3, 11) is in the row space of A

Answer 1

r1 = [1, 0, -3, 8]
r2 = [0,1,5,7]

is the basis for the rowspace of A. I am not sure how to determine if v is in the row space of A?

Answer 2

Can you get v as a linear combination of r1 and r2?

Answer 3

so basically if v = x[r1] + y[r2] and you can solve for x and y, then v is in the row space?

Answer 4

Yes. The row space is the set of vectors that you can get that way.

Answer 5

I'll be honest here I am still stuck and not sure you determine if V is in the row space.

Answer 6

The trick from what I get is that the row for every pivot point of the matrix will be a row space.

Answer 7

make a matrix with rows r1, r2, and v
rref(A) if you get 3 pivots, then v is not in the row space.

Answer 8

You basically follow the same path when trying to find the basis for the Col A.

Answer 9

as you said, we want to see if we can solve x*r1+y*r2=v
x and y are real numbers. x*r1 and y*r2 are vectors.

We multiply a vector by a real number by multiplying every component.

x*r1=(x, 0, -3x, -8x)
y*r2=(0, y, 5y, 7y)

Then add them together

x*r1+y*r2=(x, y, -x+5y, -8x+7y)

this should be equal to (4, 3, 3, 11). Two vectors are equal if all their components are equal. From the first two components we get

x=4
y=3

Substituting into x*r1+y*r2 we get (4, 3, 11, -10). So we can't possibly chose numbers x, y such that x*r1+y*r2=v

So we can't get v as a linear combination of r1 and r2. In other words v isn't in the rowspace of r1 and r2.

Answer 10

"is the basis for the rowspace of A."
The basis for the row space includes all the rowspaces of the matrix A.
That means r1 and r2 are your only row spaces for A
and if they don't equal to either vector v then vector v is not a rowspace.

Answer 11

"make a matrix with rows r1, r2, and v rref(A) if you get 3 pivots, then v is not in the row space."

I think these kinds of questions are sometimes asked before row reduction is taught.

Answer 12

@beginnersmind I don't think so. Row reduction is the first thing I learn. It's in chapter one of my book.

Answer 13

Still doesn't work, but I think "Substituting into x*r1+y*r2 we get (4, 3, 11, -10)" should be (4,3,3,53), no?

Answer 14

If it doesn't work then it's not a rowspace.

Answer 15

Ok, still my solution works too :)

Answer 16

can someone clarify a pivot point?

Answer 17

here's what I'm assuming is correct?
put v, r1, and r2 in a matrix. IF I get the v row to equal zero through elementary row operations, then that would mean v is a linear combination of r1 and r2. is this correct?

Answer 18

yes, that's right

Answer 19

ok, well my apologies, I mis-posted my row space.

v = 4 3 3 11
r1 = 1 0 -3 8
r2 = 0 1 5 -7

in rref, v = 0 0 0 0 therefore v IS IN the rowspace of A.

thanks again everyone for your help.

Answer 20

Your original post:
"dranobob

r1 = [1, 0, -3, 8] r2 = [0,1,5,7!!!!]"

Which was the correct one?

No problem though, the point is getting the method right, not the actual result.