Linear Algebra Answers

T: R--->R defined by T(x) = x+1, for all x ε R. Is T is

linear?

T: R--->R defined by T(x) = x+1, for all x ε R. Is T is

linear?

Suppose u, v "\\in" V and ||u|| = ||v|| = 1 with < u,v > = 1: Prove that u = v.

Suppose u, v "\\in" V. Prove that ||au+bv|| = ||bu+av|| for all a, b "\\in" R if and only if ||u|| = ||v||.

Suppose T is the element of L(V ) and dim range T = k. Prove that T has at

most k + 1 distinct eigenvalues.

Find vectors u,v is the element of R² such that u is a scalar multiple of (1,3), v is orthogonal to (1,3), and (1,2) = u + v.

Suppose u, v is the element of V and || u || = || v || = 1 with < u,v > = 1: Prove

that u = v.

Suppose u; v is the element of V . Prove that || au + bv || = || bu + av || for all a, b is the element of R if and only if || u || = || v || .

Suppose T "\\isin" L(V) is invertible.

(a) Suppose "\\lambda\\isin" F with "\\lambda" not= 0. Prove that "\\lambda" is an eigenvalue of T if and only if 1/"\\lambda" is an eigenvalue of T^-1.

(b) Prove that T and T^-1 have the same eigenvectors.

Suppose V is finite-dimensional and S,T "\\isin" L (V). Prove that ST and TS have the same eigenvalues