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Question 1

## Homework Statement

Find the basis of {(x,y,z) | x + y + 2z = 0}

## Homework Equations

None?

## The Attempt at a Solution

I can find the basis of a matrix but I'm not sure how to go about finding the matrix of a homogeneous equation. Can I just turn it into a 1x3 matrix, give it the coefficients (1,1,2) and call that my basis?

Question 2

## Homework Statement

Let T(y) = y'' + y. Find ker(T)

## Homework Equations

ker(T) = nullspace(matrix(T))

solution to y + y'' = 0 is asin(x) + bcos(x) (Relevant?)

## The Attempt at a Solution

I'm not even sure how to begin this. I know what the kernel of a linear transformation is but I am not sure how to turn y + y'' into a matrix. Could I do as follows?

y y' y''

1 0 1

[1, 0, 1 | 0] = REF [1, 0, 1 | 0]

y = -y''

Would that be the solution?

Finally, I have a quick question that doesn't really warrant the entire template. I'm to calculate an eigenvector and after plugging one of the eigenvalues I found into the matrix, I got

0 2 0

0 1 3

0 0 -1

Row reduction gives

0 1 3

0 0 1

0 0 0

Setting the matrix equal to 0:

x3 = 0

x2 = -3(x3) = 0

x1 = t?

In this case, is my eigenvector (t,0, 0)?

Sorry for all the questions; the exam is on Monday and the prof didn't give out any answers to his study materials. I just got around to these questions so I would really like to figure them out before the exam. Thanks for any help or advice.