Consider the matrix A = [[2, 1, 3],[4, 1, 4]]. Find an equation for the row space R(A) of A.
The row space of `A` is the set of all possible linear combinations of the rows of `A` .
Let `A = (overline(r_1),overline(r_2))`
Then the row space of `A` can be written as
`R(A) = c_1overline(r_1) + c_2overline(r_2)`
where `c_1` and `c_2` are scalar variables.
The row space is not affected by elementary row reductions. Reducing `A` we get
`A = ([2,1,3],[4,1,4]) = ([1,1/2,3/2],[4,1,4]) = ([1,1/2,3/2],[0,1,2]) = ([1,0,1/2],[0,1,2])` .
`R(A) = c_1(1,0,1/2) + c_2(0,1,2) = (c_1,c_2,1/2c_1+2c_2 )`
the set of all vectors `(x,y,z) in RR^3` such that
`z = 1/2x + 2y`
R(A) : z = 1/2x + 2y
This is a line in 3D space
Given matrix is `A=[[2,1,3],[4,1,4]]` .
As there are three columns in the given matrix. so, we need three variables `x_1,x_2,x_3` to express the row space of the matrix A.
Now we can write
`[[2,1,3],[4,1,4]]` `~~[[1,1/2,3/2],[4,1,4]]` `~~[[1,1/2,3/2],[0,1,2]]`
So, we can write the above matrix in equation form as
`and ` `x_2+2x_3=0` .
Which is the desired system of equation for the row space of the matrix A.