14.3 Find a basis for NS(A) and dim{NS(A)}

[karush]

For the matrix
$A=\left[\begin{array}{rrrrr}
1&0&0&4&5\\
0&1&0&3&2\\
0&0&1&3&2\\
0&0&0&0&0\end{array}\right]$
Find a basis for NS(A) and $\dim{NS(A)}$
$\left[\begin{array}{c}
x_1 \\
x_2 \\
x_3 \\
x_4 \\
x_5
\end{array}\right]=
\left[\begin{array}{c}
-4x_4-5x_5\\
-3x_4-2x_5\\
-3x_4-2x_5\\
x_4\\
x_5
\end{array}\right]$

ok I just did this but there is duplication in it

 

[HallsofIvy]

"NS(A)" is the null space? If so then we are looking for \begin{bmatrix}x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \end{bmatrix} such that \begin{bmatrix}1 & 0 & 0 & 4 & 5 \\ 0 & 1 & 0 & 3 & 2 \\ 0 & 0 & 1 & 3 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}\begin{bmatrix}x_1 \\ x_2 \\ x_3 \\x_4 \\ x_5\end{bmatrix}= \begin{bmatrix}x_1+ 4x_4+ 5x_5 \\ x_2+ 3x_4+ 2x_5 \\ x_3+ 3x_4+ 2x_5 \\ 0 \end{bmatrix}= \begin{bmatrix}0 \\ 0 \\ 0 \\ 0 \end{bmatrix}.

Although I wouldn't have written the equations this way they do give, as you say, x_1= -4x_4- 5x_5, x_2= -3x_4- 2x_5, x_3= -3x_4- 2x_5, and 0= 0. If by "duplication" you mean x_2= -3x_4- 2x_5 and x_3= -3x_4- 2x_5, that just means that x_2= x_3 Since all of x_1, x_2, and x_3 depend upon x_4 and x_5 take them as parameters (and the null space is two dimensional).

In particular, taking x_4= 1 and x_5= 0, x_1= -4. x_2= -3, and x_3= -3. One vector in the null space is \begin{bmatrix}-4 \\ -3 \\ -3 \\ 1 \\ 0 \end{bmatrix}. Taking x_4= 0 and x_5= 1, x_1= -5, x_2= -2, and x_3= -2. Another vector in the null space is \begin{bmatrix}-5 \\ -2 \\ -2 \\ 0 \\ 1\end{bmatrix}. Since the null space is two dimensional and the those vectors are independent, they form a basis for the null space.

 

[karush]

$\left[ \begin{array}{c}
- 5x_4 - 4x_5 \\ - 2x_4 - 3x_5\\ - 2x_4 - 3x_5 \\x_4 \\x_5
\end{array} \right]
=\left[ \begin{array}{r} -4 \\-3 \\ -3 \\ 1 \\0
\end{array} \right]x_4
+\left[ \begin{array}{r} -5 \\ -2 \\ -2 \\ 0 \\ 1
\end{array} \right]x_5$
the basis for the null space is
$\left[ \begin{array}{r} -4 \\-3 \\ -3 \\ 1 \\0
\end{array} \right]
,\left[ \begin{array}{r} -5 \\ -2 \\ -2 \\ 0 \\ 1
\end{array} \right]$

kinda getin it