Find the Kernel

$S ([\begin{array}{c} a \\ b \\ c \end{array}]) = [\begin{array}{c} a - 2 b - c \\ 3 a - b + 2 c \\ a + b + 2 c \end{array}]$

Step 1

The kernel of a transformation is a vector that makes the transformation equal to the zero vector (the pre-image of the transformation).

$[\begin{array}{c} a - 2 b - c \\ 3 a - b + 2 c \\ a + b + 2 c \end{array}] = 0$

Step 2

Create a system of equations from the vector equation.

$a - 2 b - c = 0$

$3 a - b + 2 c = 0$

$a + b + 2 c = 0$

Step 3

Write the system as a matrix.

$[\begin{array}{c} 1 & - 2 & - 1 & 0 \\ 3 & - 1 & 2 & 0 \\ 1 & 1 & 2 & 0 \end{array}]$

Step 4

Find the reduced row echelon form.

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Step 4.1

Perform the row operation $R_{2} = R_{2} - 3 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 4.1.1

Perform the row operation $R_{2} = R_{2} - 3 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - 2 & - 1 & 0 \\ 3 - 3 \cdot 1 & - 1 - 3 \cdot - 2 & 2 - 3 \cdot - 1 & 0 - 3 \cdot 0 \\ 1 & 1 & 2 & 0 \end{array}]$

Step 4.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & - 1 & 0 \\ 0 & 5 & 5 & 0 \\ 1 & 1 & 2 & 0 \end{array}]$

Step 4.2

Perform the row operation $R_{3} = R_{3} - R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 4.2.1

Perform the row operation $R_{3} = R_{3} - R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - 2 & - 1 & 0 \\ 0 & 5 & 5 & 0 \\ 1 - 1 & 1 + 2 & 2 + 1 & 0 - 0 \end{array}]$

Step 4.2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & - 1 & 0 \\ 0 & 5 & 5 & 0 \\ 0 & 3 & 3 & 0 \end{array}]$

Step 4.3

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

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Step 4.3.1

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 2 & - 1 & 0 \\ \frac{0}{5} & \frac{5}{5} & \frac{5}{5} & \frac{0}{5} \\ 0 & 3 & 3 & 0 \end{array}]$

Step 4.3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & - 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 3 & 3 & 0 \end{array}]$

Step 4.4

Perform the row operation $R_{3} = R_{3} - 3 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 4.4.1

Perform the row operation $R_{3} = R_{3} - 3 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 2 & - 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 - 3 \cdot 0 & 3 - 3 \cdot 1 & 3 - 3 \cdot 1 & 0 - 3 \cdot 0 \end{array}]$

Step 4.4.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & - 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 4.5

Perform the row operation $R_{1} = R_{1} + 2 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 4.5.1

Perform the row operation $R_{1} = R_{1} + 2 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & - 1 + 2 \cdot 1 & 0 + 2 \cdot 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 4.5.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 5

Use the result matrix to declare the final solution to the system of equations.

$a + c = 0$

$b + c = 0$

$0 = 0$

Step 6

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} a \\ b \\ c \end{array}] = [\begin{array}{c} - c \\ - c \\ c \end{array}]$

Step 7

Write the solution as a linear combination of vectors.

$[\begin{array}{c} a \\ b \\ c \end{array}] = c [\begin{array}{c} - 1 \\ - 1 \\ 1 \end{array}]$

Step 8

Write as a solution set.

${c [\begin{array}{c} - 1 \\ - 1 \\ 1 \end{array}] | c \in R}$

Step 9

The solution is the set of vectors created from the free variables of the system.

${[\begin{array}{c} - 1 \\ - 1 \\ 1 \end{array}]}$

Step 10

The kernel of $S$ is the subspace ${[\begin{array}{c} - 1 \\ - 1 \\ 1 \end{array}]}$ .

$K (S) = {[\begin{array}{c} - 1 \\ - 1 \\ 1 \end{array}]}$

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