So I got this problem in which I had to determine whether a vector [1,2,5,-1] was in the row space of the following matrix:
First row: 2, -1, 0, 3
2nd row: 7, -1, 5, 8
I know that in order to do this problem, you have to set up the matrix:
2 7 |1
-1 -1 | 2
0 5 | 5
3 8 | - 1
I row reduced it using row-reduced echelon form, and I was wondering if I got a correct answer (my graphic calculator doesn"t work for some reason when I create matrices in which the number of rows exceed the number of columns so I couldn't use it to check my work).
Using Type I (what you use to turn a pivot entry to 1) and Type II (what you use to turn a number into 0, which means taking the negative of the desired entry, multiplying it by the pivot row, and adding the row with the desired entry) row operations, I finally wound up with the matrix:
1 0 | 5/3
0 1 | -1/3
0 0 | 20/3
0 0 | -10/3
I have been thinking and I have another way of doing this.
First note that the two given vectors are linearly independent (why?). Thus, they form a real vector space of dimension 2.
Now, [1,2,5,-1] is in the row space of two two vectors if and only if the row space of all three is the same.
Meaning, show that,
[1 2 5 -1]
[2 -1 0 3]
[7 -1 5 8]
Has rank = 2.
okay, so i realized there was an error in calculating my matrix in my first post.
when i put the matrix
{2, 7 | 1}
{-1, -1 | 2}
{0, 5 | 5}
{3, 8 | -1}
in row reduced echelon form, I got
{1, 0 | 3/2}
{0, 1 | 3/5}
{0, 0 | 2 }
{0, 0 | -1}
Therefore, the rank of this matrix is 2 because there are two rows with nonzero pivot entries. But how does one go about finding the A and the B for linear combination?
Since you where working with the coloum spaces you can see that that actual vectors are the corresponding colums in the original matrix.
By looking note that the third coloum is 3/5 the second and 3/2 the first.
That means those constants express that vector in terms of those ones (in the corresponding matrix).
Okay, so A is equal to 3/2 and B is equal to 3/5?
If this is true, then shouldn't 3/2{2, -1, 0, 3) + 3/5 (7, -1, 5, 8) yield the the vector (1, 2, 5, -1)?
Because when I do this out, I get a different answer.