Find the Kernel 4a+3b=[[6,5],[1,9]][[9,1],[4,5]]

Problem

4*a+3*b=[[6,5],[1,9]]*[[9,1],[4,5]]

Solution

1. Multiply the two matrices on the right side of the equation.

[[6,5],[1,9]]*[[9,1],[4,5]]=[[(6⋅9+5⋅4),(6⋅1+5⋅5)],[(1⋅9+9⋅4),(1⋅1+9⋅5)]]

[[54+20,6+25],[9+36,1+45]]=[[74,31],[45,46]]

2. Set up the equation for the linear combination of variables a and b

4*a+3*b=[[74,31],[45,46]]

3. Identify the task requirement. The "Kernel" (or null space) of a linear transformation T(v)=0 refers to the set of all vectors that map to the zero element. However, the provided expression is an equation of the form L(a,b)=M where M is a constant matrix.

4. Determine the kernel of the underlying linear operator L(a,b)=4*a+3*b For the kernel, we solve the homogeneous equation where the output is the zero matrix.

4*a+3*b=[[0,0],[0,0]]

5. Solve for one variable in terms of the other to find the general form of the vectors in the kernel.

3*b=−4*a

b=−4/3*a

6. Express the kernel as a set of pairs (a,b) where a is any matrix in the same space (presumably 2×2 matrices).

Ker(L)={(a,−4/3*a):a∈(𝕄_2×2)}

Final Answer

Ker(4*a+3*b)={(a,b):4*a+3*b=0}⇒b=−4/3*a

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