proof

Please do NOT send the answer back as an attachment. Please show work. Thank you. Carefully write a proof for the following statement: Similar matrices have the same characteristic polynomials.

proof of eigenvalue

Please do NOT send the answer back as an attachment. Please show work. Thank you. Carefully write a proof for the following statement: Any n x n singular matrix A must have zero as an eigenvalue.

Linear algebra and differential equations

Determine if the given set constitutes a real vector space. The operations of "multiplication by a number" and "addition" are understood to be the usual operations associated with the elements of the set: The set of all elements of R^3 with first component 0

linear algebra and differential equations

Please show all work; don't explain each step. Please DON'T submit back as an attachment.Thank you. Determine if the given set constitutes a real vector space. The operations of "multiplication by a number" and "addition" are understood to be the usual operations associated with the elements of the set: The set of all polynomials of degree  2

linear algebra

Please show all work; don't explain each step. Please DON'T submit back as an attachment.Thank you. Show that the set of all elements of R^2 of the form (a, -a), where a is any real number, is a subspace of R^2. Give a geometric interpretation of the subspace.