How to solve this to find the Null Space

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What I did:

I put this into reduced row echelon form:

$$\begin{bmatrix} 1 & -2 & 2 & 4 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$

It is clear that the $r(M)=2$, because there are two independent rows.


Now for the null space, I wrote down the equations from the reduced row echelon form:

$$x-2y+2z+4t=0$$

$$z+t=0$$

I can’t seem to write $x$ and $y$ separately in terms of $z$ and $t$. Any hints?

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Write the system

$$\left\{\begin{array}{rcl} x-2y+2z+4t& =& 0 \\ z+t & = & 0\end{array}\right.$$

as

$$\left\{\begin{array}{rcl} x+2z& =& 2y-4t \\ z & = & -t\end{array}\right.$$ and solve it. You get,

$$\left\{\begin{array}{rcl} x& =& 2y-2t \\ z & = & -t\end{array}\right.$$ That is, $$(2y-2t,y,-t,t)$$ is an element of the null space for any $y,t.$ Now, look for two linearly independent vectors.

(Note that the kernel has dimension $2.$ So the system has infinitely many solutions that have to depend on two parameters.)

Do also backwards elimination:
$$
\begin{bmatrix}
1 & -2 & 2 & 4 \\
0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}
\to
\begin{bmatrix}
1 & -2 & 0 & 2 \\
0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}
$$
Now your equations read
$$
\begin{cases}
x_1=2x_2-2x_4\\
x_3=-x_4
\end{cases}
$$
You get two linearly independent vectors in the null space by setting $x_2=1, x_4=0$ and $x_2=0,x_4=1$, so the basis is given by the two linearly independent vectors
$$
\begin{bmatrix}
2\\
1\\
0\\
0\\
\end{bmatrix}
\qquad
\begin{bmatrix}
-2\\
0\\
-1\\
1
\end{bmatrix}
$$
The first corresponds to $x_2=1$ and $x_4=0$, the second to $x_2=0$ and $x_4=1$.

Just to make the answer a bit more algorithmic: a “pivot entry” is an entry which is the first non-zero entry in its row. A “pivot column” is a column containing a pivot entry. In your matrix, columns 1 and 3 are pivot columns.

Name the variables after the columns as you did (so, $x, y, z, w$). Then the “free variables” are the ones that don’t occur in pivot columns. In your case, these are $y$ and $w$. The remaining rows of the matrix express the bound variables in terms of the free variables.

First case: $z=t=0$, you obtain $x-2y=0$. One of solutions is the vector $(2,1,0,0)$.

Second case: $z=-t=1$, which gives you the equation $x-2y=2$, which gives you, for example, $(2,0,1,-1)$.

Write $z = -t$ and put it in your first equation. You shall get $x = 2y + 6t$, after simplification. Write $$(x,y,z,t)’ = y(2 , 1 , 0 , 0)’ + t(6, 0, -1, 1)’$$. Now consider $y$ and $t$ in some field. This shall give you all the solutions.