Simplify the Matrix [[8b-2c],[-b],[7c]]

Problem

[[8*b−2*c],[−b],[7*c]]

Solution

1. Identify the structure of the matrix, which is a 3×1 column vector where each entry is a linear combination of the variables b and c

2. Decompose the matrix into the sum of two separate column vectors, one containing only terms with b and the other containing only terms with c

[[8*b−2*c],[−b],[7*c]]=[[8*b],[−b],[0]]+[[−2*c],[0],[7*c]]

3. Factor out the scalar variables b and c from their respective vectors to express the matrix as a linear combination of constant vectors.

b*[[8],[−1],[0]]+c*[[−2],[0],[7]]

Final Answer

[[8*b−2*c],[−b],[7*c]]=b*[[8],[−1],[0]]+c*[[−2],[0],[7]]

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