Find bases for the four fundamental subspaces of the matrix...

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00:01 Hello again.

00:03 Good to see you.

00:04 All right.

00:05 So for this particular problem, we are to find the four fundamental subspaces of a given matrix.

00:15 And the four fundamental subspaces are the row space, the column space, the null space, and the left null space.

00:25 So let's get started.

00:27 So we have our matrix as 1 is 1 0 negative 1 0 negative 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1.

00:59 So this is our matrix.

01:09 Ok, so what we're going to do is we are going to write this matrix in reduce row echelon form.

01:15 So when you manipulate and flip the different rows and you multiply, your reduced row echelon form becomes 1 -0 -0 -0 -1 -0 -0.

01:48 And 0 -0 -0.

01:54 So this is our matrix a in reduced row echelon form because we have one on the main diagonals.

02:06 Now, given that this is now in reduced row echelon form and our last row is all zeros, we can just leave it out.

02:17 So when we talk about the row space, our row space is the number of rows we have.

02:22 Okay, and we're not counting the last row because it's all zeros.

02:27 So therefore, our row space is going to be three.

02:43 Okay, and our column space stays the same because we didn't get rid of any columns...

Find bases for the four fundamental subspaces of the matrix $A$. $$A=\left[\begin{array}{rrr} 1 & 0 & -1 \\ 0 & -1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{array}\right]$$