Project Using the Transformation A=[[5,3,-2],[0,4,-1]] , x=[[3],[1],[2]]

Problem

A=[[5,3,−2],[0,4,−1]],x=[[3],[1],[2]]

Solution

1. Identify the operation as a matrix-vector multiplication A*x to project the vector x using the transformation matrix A

2. Set up the multiplication by placing the matrix A and the vector x in the standard form A*x

A*x=[[5,3,−2],[0,4,−1]]*[[3],[1],[2]]

3. Calculate the first component of the resulting vector by taking the dot product of the first row of A and the vector x

(5)*(3)+(3)*(1)+(−2)*(2)=15+3−4=14

4. Calculate the second component of the resulting vector by taking the dot product of the second row of A and the vector x

(0)*(3)+(4)*(1)+(−1)*(2)=0+4−2=2

5. Combine the results into a new vector.

[[14],[2]]

Final Answer

[[5,3,−2],[0,4,−1]]*[[3],[1],[2]]=[[14],[2]]

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