Find the basis for kernel of a matrix transformation Let ψ :{Mat}2×2(R) → {Mat}2×2(R) ψ:(abcd)→(a+ba−ca+cb−c) Find basis for ker ψ

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Find the basis for kernel of a matrix transformation  Let ψ :{Mat}2×2(R) → {Mat}2×2(R)  ψ:(abcd)→(a+ba−ca+cb−c)  Find basis for ker ψ

trasahed · Accepted Answer

Step 1Assuming that Mat2×2(R) are being though of as additive groups, the kernel is made up of all matrices that are send to the zero matrix.So you need a+b=0, a−c=0, a+c=0 and b−c=0. Well, a+b=0 if and only if a=−b. Let&#039;s substitute that into the remaining three equations: −b−c=0, −b+c=0 and b−c=0. Well, −b−c=0 if and only if b=−c. Let&#039;s put that into the two remaining equations: −(−c)+c=0 and −(−c)−c=0. Well, 2c=0 if and only if c=0. Hence a=−b, b=−c and c=0 meaning that a=b=c=0.You&#039;re free to choose d. So the matrix[0001]spans the kernel.

Kyler Jacobson · Accepted Answer

Step 1  First identify Mat2×2 with R4, with basis matrices  (1000),⋯,  and write ψ as a linear transformation R4→R4 with matrix M.   Then read the examples in this Wikipedia article, which give a method for finding a basis once you have a matrix.

Find the basis for kernel of a matrix transformation Let \psi\

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