The first question “The vector b is in the kernel of T” I got as false because I did A*b and got (-7, 24, -25) which is not a 0 vector so I put it as false.
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The second question “The vector c is in the range of T” I put as false because I was unable to do an augmented matrix of Ax = c because the dimensions are not the same.
I got the question right because they both turned out to be false but I am not sure if my reasoning are correct. Can somebody verify if I am interpreting this problem correctly? Thank you
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Both are correct reasoning. The kernel of a linear transformation A is by definition the set of all vectors which get mapped to 0; it is as simple as calculating Ab and comparing it to 0 to see whether or not b is in the kernel. The vector c can only be in the range of A only if it lies in the codomain of A, which is R3 for your example; as c lies in R2, it couldn't possibly be in the range of A.
Op · 7y · edited 7y
Okay, thank you. I just wanted to make sure I understood the concepts for the upcoming exam.
Edit: If the c vector was in the same dimensions would that automatically make the vector in the range? Or would I have to do RREF to determine if it's in the range?
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