Linear Algebra Examples

$[\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} y \\ x \end{array}]$

Step 1

The transformation defines a map from $ℝ^{2}$ to $ℝ^{2}$ . To prove the transformation is linear, the transformation must preserve scalar multiplication, addition, and the zero vector.

M: $ℝ^{2} \to ℝ^{2}$

Step 2

First prove the transform preserves this property.

$M (x + y) = M (x) + M (y)$

Step 3

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

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

Step 4

Add the two matrices.

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

Step 5

Apply the transformation to the vector.

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

Step 6

Break the result into two matrices by grouping the variables.

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

Step 7

The addition property of the transformation holds true.

$M (x + y) = M (x) + M (y)$

Step 8

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

$M (p x) = T (p [\begin{array}{c} x \\ y \end{array}])$

Step 9

Factor the $p$ from each element.

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Step 9.1

Multiply $p$ by each element in the matrix.

$M (p x) = M ([\begin{array}{c} p x \\ p y \end{array}])$

Step 9.2

Apply the transformation to the vector.

Step 9.3

Rearrange $p y$ .

$M (p x) = [\begin{array}{c} p y \end{array}]$

Step 9.4

Factor element $0, 0$ by multiplying $p y$ .

$M (p x) = [\begin{array}{c} p (y) \end{array}]$

Step 10

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

$M (p [\begin{array}{c} x \\ y \end{array}]) = p M (x)$

Step 11

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

$M (0) = 0$

Step 12

Apply the transformation to the vector.

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

Step 13

Simplify each element in the matrix.

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Step 13.1

Rearrange $0$ .

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

Step 13.2

Rearrange $0$ .

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

Step 14

The zero vector is preserved by the transformation.

$M (0) = 0$

Step 15

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

Linear Transformation