Linear Algebra Examples

$[\begin{array}{c} 2 & 3 \\ 1 & 4 \end{array}] A = [\begin{array}{c} 3 & 0 \\ 1 & 2 \end{array}]$

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.

$M (x + y) = [\begin{array}{c} 3 \\ 1 \end{array}]$

Step 6

Break the result into two matrices by grouping the variables.

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

Step 7

Since the transformation addition property does not hold, this is not a linear transformation.

$M (x + y) \neq M (x) + M (y)$