Linear Algebra Examples

S ⎛ ⎜ ⎝ ⎡ ⎢ ⎣ \begin{matrix} a b c \end{matrix} ⎤ ⎥ ⎦ ⎞ ⎟ ⎠ = ⎡ ⎢ ⎣ \begin{matrix} a - 6 b - 3 c a - 2 b + c a + 3 b + 5 c \end{matrix} ⎤ ⎥ ⎦

$S ([\begin{array}{c} a \\ b \\ c \end{array}]) = [\begin{array}{c} a - 6 b - 3 c \\ a - 2 b + c \\ a + 3 b + 5 c \end{array}]$

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{matrix} a - 6 b - 3 c a - 2 b + c a + 3 b + 5 c \end{matrix} ⎤ ⎥ ⎦ = 0

$[\begin{array}{c} a - 6 b - 3 c \\ a - 2 b + c \\ a + 3 b + 5 c \end{array}] = 0$

Step 2

Create a system of equations from the vector equation.

a - 6 b - 3 c = 0

$a - 6 b - 3 c = 0$

a - 2 b + c = 0

$a - 2 b + c = 0$

a + 3 b + 5 c = 0

$a + 3 b + 5 c = 0$

Step 3

Write the system as a matrix.

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 1 & - 2 & 1 & 0 1 & 3 & 5 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ 1 & - 2 & 1 & 0 \\ 1 & 3 & 5 & 0 \end{array}]$

Step 4

Find the reduced row echelon form.

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Step 4.1

Perform the row operation

R_{2} = R_{2} - R_{1}

$R_{2} = R_{2} - R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Step 4.1.1

Perform the row operation

R_{2} = R_{2} - R_{1}

$R_{2} = R_{2} - R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 1 - 1 & - 2 + 6 & 1 + 3 & 0 - 0 1 & 3 & 5 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ 1 - 1 & - 2 + 6 & 1 + 3 & 0 - 0 \\ 1 & 3 & 5 & 0 \end{array}]$

Step 4.1.2

Simplify

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 0 & 4 & 4 & 0 1 & 3 & 5 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ 0 & 4 & 4 & 0 \\ 1 & 3 & 5 & 0 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 0 & 4 & 4 & 0 1 & 3 & 5 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ 0 & 4 & 4 & 0 \\ 1 & 3 & 5 & 0 \end{array}]$

Step 4.2

Perform the row operation

R_{3} = R_{3} - R_{1}

$R_{3} = R_{3} - R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

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Step 4.2.1

Perform the row operation

R_{3} = R_{3} - R_{1}

$R_{3} = R_{3} - R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 0 & 4 & 4 & 0 1 - 1 & 3 + 6 & 5 + 3 & 0 - 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ 0 & 4 & 4 & 0 \\ 1 - 1 & 3 + 6 & 5 + 3 & 0 - 0 \end{array}]$

Step 4.2.2

Simplify

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 0 & 4 & 4 & 0 0 & 9 & 8 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ 0 & 4 & 4 & 0 \\ 0 & 9 & 8 & 0 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 0 & 4 & 4 & 0 0 & 9 & 8 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ 0 & 4 & 4 & 0 \\ 0 & 9 & 8 & 0 \end{array}]$

Step 4.3

Multiply each element of

R_{2}

$R_{2}$ by

\frac{1}{4}

$\frac{1}{4}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Step 4.3.1

Multiply each element of

R_{2}

$R_{2}$ by

\frac{1}{4}

$\frac{1}{4}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 \frac{0}{4} & \frac{4}{4} & \frac{4}{4} & \frac{0}{4} 0 & 9 & 8 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ \frac{0}{4} & \frac{4}{4} & \frac{4}{4} & \frac{0}{4} \\ 0 & 9 & 8 & 0 \end{array}]$

Step 4.3.2

Simplify

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 0 & 1 & 1 & 0 0 & 9 & 8 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 9 & 8 & 0 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 0 & 1 & 1 & 0 0 & 9 & 8 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 9 & 8 & 0 \end{array}]$

Step 4.4

Perform the row operation

R_{3} = R_{3} - 9 R_{2}

$R_{3} = R_{3} - 9 R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

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Step 4.4.1

Perform the row operation

R_{3} = R_{3} - 9 R_{2}

$R_{3} = R_{3} - 9 R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 0 & 1 & 1 & 0 0 - 9 \cdot 0 & 9 - 9 \cdot 1 & 8 - 9 \cdot 1 & 0 - 9 \cdot 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ 0 & 1 & 1 & 0 \\ 0 - 9 \cdot 0 & 9 - 9 \cdot 1 & 8 - 9 \cdot 1 & 0 - 9 \cdot 0 \end{array}]$

Step 4.4.2

Simplify

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 0 & 1 & 1 & 0 0 & 0 & - 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & - 1 & 0 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 0 & 1 & 1 & 0 0 & 0 & - 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & - 1 & 0 \end{array}]$

Step 4.5

Multiply each element of

R_{3}

$R_{3}$ by

- 1

$- 1$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

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Step 4.5.1

Multiply each element of

R_{3}

$R_{3}$ by

- 1

$- 1$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 0 & 1 & 1 & 0 - 0 & - 0 & - - 1 & - 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ 0 & 1 & 1 & 0 \\ - 0 & - 0 & - - 1 & - 0 \end{array}]$

Step 4.5.2

Simplify

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 0 & 1 & 1 & 0 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 0 & 1 & 1 & 0 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Step 4.6

Perform the row operation

R_{2} = R_{2} - R_{3}

$R_{2} = R_{2} - R_{3}$ to make the entry at

2, 3

$2, 3$ a

0

$0$ .

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Step 4.6.1

Perform the row operation

R_{2} = R_{2} - R_{3}

$R_{2} = R_{2} - R_{3}$ to make the entry at

2, 3

$2, 3$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 0 - 0 & 1 - 0 & 1 - 1 & 0 - 0 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ 0 - 0 & 1 - 0 & 1 - 1 & 0 - 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Step 4.6.2

Simplify

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 0 & 1 & 0 & 0 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & - 3 & 0 0 & 1 & 0 & 0 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & - 3 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Step 4.7

Perform the row operation

R_{1} = R_{1} + 3 R_{3}

$R_{1} = R_{1} + 3 R_{3}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

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Step 4.7.1

Perform the row operation

R_{1} = R_{1} + 3 R_{3}

$R_{1} = R_{1} + 3 R_{3}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 + 3 \cdot 0 & - 6 + 3 \cdot 0 & - 3 + 3 \cdot 1 & 0 + 3 \cdot 0 0 & 1 & 0 & 0 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 + 3 \cdot 0 & - 6 + 3 \cdot 0 & - 3 + 3 \cdot 1 & 0 + 3 \cdot 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Step 4.7.2

Simplify

R_{1}

$R_{1}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & 0 & 0 0 & 1 & 0 & 0 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 6 & 0 & 0 0 & 1 & 0 & 0 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & - 6 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Step 4.8

Perform the row operation

R_{1} = R_{1} + 6 R_{2}

$R_{1} = R_{1} + 6 R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

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Step 4.8.1

Perform the row operation

R_{1} = R_{1} + 6 R_{2}

$R_{1} = R_{1} + 6 R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 + 6 \cdot 0 & - 6 + 6 \cdot 1 & 0 + 6 \cdot 0 & 0 + 6 \cdot 0 0 & 1 & 0 & 0 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 + 6 \cdot 0 & - 6 + 6 \cdot 1 & 0 + 6 \cdot 0 & 0 + 6 \cdot 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Step 4.8.2

Simplify

R_{1}

$R_{1}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 & 0 0 & 1 & 0 & 0 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

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

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 & 0 0 & 1 & 0 & 0 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

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

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 & 0 0 & 1 & 0 & 0 0 & 0 & 1 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

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

Step 5

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

a = 0

$a = 0$

b = 0

$b = 0$

c = 0

$c = 0$

Step 6

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

⎡ ⎢ ⎣ \begin{matrix} a b c \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 000 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} a \\ b \\ c \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 7

Write as a solution set.

⎧ ⎪ ⎨ ⎪ ⎩ ⎡ ⎢ ⎣ \begin{matrix} 000 \end{matrix} ⎤ ⎥ ⎦ ⎫ ⎪ ⎬ ⎪ ⎭

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

Step 8

The kernel of

$S$ is the subspace

${[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]}$ .

$K (S) = {[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]}$

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