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

FiessyFrimatsd0

FiessyFrimatsd0

Answered question

2022-01-29

Find the basis for kernel of a matrix transformation
Let ψ :{Mat}2×2(R)  {Mat}2×2(R)
ψ:(abcd)(a+baca+cbc)
Find basis for ker ψ

Answer & Explanation

trasahed

trasahed

Beginner2022-01-30Added 14 answers

Step 1
Assuming 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, ac=0, a+c=0 and bc=0. Well, a+b=0 if and only if a=b. Let's substitute that into the remaining three equations: bc=0, b+c=0 and bc=0. Well, bc=0 if and only if b=c. Let'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're free to choose d. So the matrix
[0001]
spans the kernel.
Kyler Jacobson

Kyler Jacobson

Beginner2022-01-31Added 8 answers

Step 1
First identify Mat2×2 with R4, with basis matrices
(1000),,
and write ψ as a linear transformation R4R4 with matrix M.
Then read the examples in this Wikipedia article, which give a method for finding a basis once you have a matrix.

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