Linear Algebra Examples

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

Step 1

The transformation defines a map from $ℝ^{3}$ to $ℝ^{3}$ . To prove the transformation is linear, the transformation must preserve scalar multiplication, addition, and the zero vector.

S: $ℝ^{3} \to ℝ^{3}$

Step 2

First prove the transform preserves this property.

$S (x + y) = S (x) + S (y)$

Step 3

Set up two matrices to test the addition property is preserved for $S$ .

$S ([\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}] + [\begin{array}{c} y_{1} \\ y_{2} \\ y_{3} \end{array}])$

Step 4

Add the two matrices.

$S [\begin{array}{c} x_{1} + y_{1} \\ x_{2} + y_{2} \\ x_{3} + y_{3} \end{array}]$

Step 5

Apply the transformation to the vector.

$S (x + y) = [\begin{array}{c} x_{1} + y_{1} - 3 (x_{2} + y_{2}) - 3 (x_{3} + y_{3}) \\ 3 (x_{1} + y_{1}) - (x_{2} + y_{2}) - 3 (x_{3} + y_{3}) \\ x_{1} + y_{1} - (x_{2} + y_{2}) + x_{3} + y_{3} \end{array}]$

Step 6

Simplify each element in the matrix.

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Step 6.1

Rearrange $x_{1} + y_{1} - 3 (x_{2} + y_{2}) - 3 (x_{3} + y_{3})$ .

$S (x + y) = [\begin{array}{c} x_{1} - 3 x_{2} - 3 x_{3} + y_{1} - 3 y_{2} - 3 y_{3} \\ 3 (x_{1} + y_{1}) - (x_{2} + y_{2}) - 3 (x_{3} + y_{3}) \\ x_{1} + y_{1} - (x_{2} + y_{2}) + x_{3} + y_{3} \end{array}]$

Step 6.2

Rearrange $3 (x_{1} + y_{1}) - (x_{2} + y_{2}) - 3 (x_{3} + y_{3})$ .

$S (x + y) = [\begin{array}{c} x_{1} - 3 x_{2} - 3 x_{3} + y_{1} - 3 y_{2} - 3 y_{3} \\ 3 x_{1} - x_{2} - 3 x_{3} + 3 y_{1} - y_{2} - 3 y_{3} \\ x_{1} + y_{1} - (x_{2} + y_{2}) + x_{3} + y_{3} \end{array}]$

Step 6.3

Rearrange $x_{1} + y_{1} - (x_{2} + y_{2}) + x_{3} + y_{3}$ .

$S (x + y) = [\begin{array}{c} x_{1} - 3 x_{2} - 3 x_{3} + y_{1} - 3 y_{2} - 3 y_{3} \\ 3 x_{1} - x_{2} - 3 x_{3} + 3 y_{1} - y_{2} - 3 y_{3} \\ x_{1} - x_{2} + x_{3} + y_{1} - y_{2} + y_{3} \end{array}]$

Step 7

Break the result into two matrices by grouping the variables.

$S (x + y) = [\begin{array}{c} x_{1} - 3 x_{2} - 3 x_{3} \\ 3 x_{1} - x_{2} - 3 x_{3} \\ x_{1} - x_{2} + x_{3} \end{array}] + [\begin{array}{c} y_{1} - 3 y_{2} - 3 y_{3} \\ 3 y_{1} - y_{2} - 3 y_{3} \\ y_{1} - y_{2} + y_{3} \end{array}]$

Step 8

The addition property of the transformation holds true.

$S (x + y) = S (x) + S (y)$

Step 9

For a transformation to be linear, it must maintain scalar multiplication.

$S (p x) = T (p [\begin{array}{c} a \\ b \\ c \end{array}])$

Step 10

Factor the $p$ from each element.

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Step 10.1

Multiply $p$ by each element in the matrix.

$S (p x) = S ([\begin{array}{c} p a \\ p b \\ p c \end{array}])$

Step 10.2

Apply the transformation to the vector.

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

Step 10.3

Simplify each element in the matrix.

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Step 10.3.1

Rearrange $(p a) - 3 (p b) - 3 (p c)$ .

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

Step 10.3.2

Rearrange $3 ((p a) - (p b) - 3 (p c))$ .

$S (p x) = [\begin{array}{c} a p - 3 b p - 3 c p \\ 3 a p - 3 b p - 9 c p \\ (p a) - (p b) + p c \end{array}]$

Step 10.3.3

Rearrange $(p a) - (p b) + p c$ .

$S (p x) = [\begin{array}{c} a p - 3 b p - 3 c p \\ 3 a p - 3 b p - 9 c p \\ a p - 1 b p + c p \end{array}]$

Step 10.4

Factor each element of the matrix.

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Step 10.4.1

Factor element $0, 0$ by multiplying $a p - 3 b p - 3 c p$ .

$S (p x) = [\begin{array}{c} p (a - 3 b - 3 c) \\ 3 a p - 3 b p - 9 c p \\ a p - 1 b p + c p \end{array}]$

Step 10.4.2

Factor element $1, 0$ by multiplying $3 a p - 3 b p - 9 c p$ .

$S (p x) = [\begin{array}{c} p (a - 3 b - 3 c) \\ p (3 a - 3 b - 9 c) \\ a p - 1 b p + c p \end{array}]$

Step 10.4.3

Factor element $2, 0$ by multiplying $a p - 1 b p + c p$ .

$S (p x) = [\begin{array}{c} p (a - 3 b - 3 c) \\ p (3 a - 3 b - 9 c) \\ p (a - b + c) \end{array}]$

Step 11

The second property of linear transformations is preserved in this transformation.

$S (p [\begin{array}{c} a \\ b \\ c \end{array}]) = p S (x)$

Step 12

For the transformation to be linear, the zero vector must be preserved.

$S (0) = 0$

Step 13

Apply the transformation to the vector.

$S (0) = [\begin{array}{c} (0) - 3 \cdot 0 - 3 \cdot 0 \\ 3 (0) - (0) - 3 \cdot 0 \\ (0) - (0) + 0 \end{array}]$

Step 14

Simplify each element in the matrix.

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Step 14.1

Rearrange $(0) - 3 \cdot 0 - 3 \cdot 0$ .

$S (0) = [\begin{array}{c} 0 \\ 3 (0) - (0) - 3 \cdot 0 \\ (0) - (0) + 0 \end{array}]$

Step 14.2

Rearrange $3 (0) - (0) - 3 \cdot 0$ .

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

Step 14.3

Rearrange $(0) - (0) + 0$ .

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

Step 15

The zero vector is preserved by the transformation.

$S (0) = 0$

Step 16

Since all three properties of linear transformations are not met, this is not a linear transformation.

Linear Transformation

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