QuestionIf is a linear transformation and the action of on the special vectors and is as given, find a formula for , where is any vector in .
Studdy Solution
STEP 1
1. is a linear transformation.
2. The vectors and are given as and .
3. The transformation acts on and as follows: and .
4. We need to find a formula for where .
STEP 2
1. Use the properties of linear transformations.
2. Express as a linear combination of and .
3. Apply the linear transformation to .
4. Simplify the expression to find the formula for .
STEP 3
Since is a linear transformation, it satisfies the properties: for any scalars and .
STEP 4
Express as a linear combination of and .
We need to find scalars and such that:
This gives us the system of equations:
STEP 5
Solve the system of equations for and .
Using the method of substitution or elimination, solve for and .
STEP 6
Let's solve the system using elimination:
Multiply the first equation by 4 and the second by 3:
Subtract the second equation from the first:
Substitute into the first equation:
STEP 7
Apply the linear transformation to using the linear combination:
Substitute and :
STEP 8
Substitute the values of and :
Calculate each component:
STEP 9
Simplify each component:
Combine like terms:
The formula for is:
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