# Consider the matrix A = [[2, 1, 3],[4, 1, 4]]. Find an equation for the row space R(A) of A.

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### 3 Answers

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])` .

Then,

`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

`[[1,1/2,3/2],[0,1,2]][[x_1],[x_2],[x_3]]=[0]`

or, `x_1+1/2x_2+3/2x_3=0`

`and ` `x_2+2x_3=0` .

or, `2x_1+x_2+3x_3=0`

and `x_2+2x_3=0.`

Which is the desired system of equation for the row space of the matrix A.

`A=[[2,1,2],[4,1,4]]`

rref of A (say) B=`[[1,0,1/2],[0,1,2]]`

Thus row space of A i.e R(A)={`[[1,0,1/2]],[[0,1,2]]` }

Ans.