Linear Algebra Examples

Step 1
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:
Step 2
First prove the transform preserves this property.
Step 3
Set up two matrices to test the addition property is preserved for .
Step 4
Add the two matrices.
Step 5
Apply the transformation to the vector.
Step 6
Simplify each element in the matrix.
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Rearrange .
Rearrange .
Rearrange .
Step 7
Break the result into two matrices by grouping the variables.
Step 8
The addition property of the transformation holds true.
Step 9
For a transformation to be linear, it must maintain scalar multiplication.
Step 10
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 in the matrix.
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Rearrange .
Rearrange .
Rearrange .
Factor each element of the matrix.
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Factor element by multiplying .
Factor element by multiplying .
Factor element by multiplying .
Step 11
The second property of linear transformations is preserved in this transformation.
Step 12
For the transformation to be linear, the zero vector must be preserved.
Step 13
Apply the transformation to the vector.
Step 14
Simplify each element in the matrix.
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Rearrange .
Rearrange .
Rearrange .
Step 15
The zero vector is preserved by the transformation.
Step 16
Since all three properties of linear transformations are not met, this is not a linear transformation.
Linear Transformation
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