Linear Algebra Examples

The transformation defines a map from to . To prove the transformation is linear, the transformation must preserve scalar multiplication, addition, and the zero vector.
S:
First prove the transform preserves this property.
Set up two matrices to test the addition property is preserved for .
Add the two matrices.
Apply the transformation to the vector.
Simplify each element of the matrix .
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Simplify element by multiplying to get .
Simplify element by multiplying to get .
Simplify element by multiplying to get .
Break the result into two matrices by grouping the variables.
The addition property of the transformation holds true.
For a transformation to be linear, it must maintain scalar multiplication.
Factor the from each element.
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Multiply by each element in the matrix.
Apply the transformation to the vector.
Simplify each element of the matrix .
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Simplify element by multiplying to get .
Simplify element by multiplying to get .
Simplify element by multiplying to get .
Factor each element of the matrix.
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Factor element by multiplying to get .
Factor element by multiplying to get .
Factor element by multiplying to get .
The second property of linear transformations is preserved in this transformation.
For the transformation to be linear, the zero vector must be preserved.
Apply the transformation to the vector.
Simplify each element of the matrix .
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Simplify element by multiplying to get .
Simplify element by multiplying to get .
Simplify element by multiplying to get .
The zero vector is preserved by the transformation.
Since all three properties of linear transformations are not met, this is not a linear transformation.
Linear Transformation
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