Find the Kernel [[a,b],[c,d]]*[[4],[1]]=[[0],[0]]

Problem

[[a,b],[c,d]]*[[4],[1]]=[[0],[0]]

Solution

1. Perform matrix multiplication on the left side of the equation by multiplying each row of the matrix by the column vector.

[[4*a+b],[4*c+d]]=[[0],[0]]

2. Set up a system of linear equations by equating the corresponding components of the resulting vectors.

4*a+b=0

4*c+d=0

3. Solve for variables b and d in terms of a and c to define the relationship between the entries of the matrix.

b=−4*a

d=−4*c

4. Express the matrix in terms of the free variables a and c to describe the set of all matrices in the kernel.

[[a,−4*a],[c,−4*c]]

5. Factor out the scalars to write the kernel as a linear combination of basis matrices.

a*[[1,−4],[0,0]]+c*[[0,0],[1,−4]]

Final Answer

Kernel={[[a,−4*a],[c,−4*c]]:a,c∈ℝ}

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