Linear Algebra Answers

Determine if the following sets are linearly dependent, or independent.

(i) {1,sin(x),cos(x)}

(ii) {sin²(x),cos(2x),cos²(x)}

Given two bases

B={1−x,2+x,3−x+x2}

and

C={1,2+x,1+x−x2}

of P2, the vector space of polynomials of degree ≤2,

(i) find p(x)∈P2 whose coordinates with respect to B is [p(x)]B=⎡⎣⎢1 −1 3⎤⎦⎥,

(ii) find the transition (change of coordinates) matrix CMB∈R3×3 from B to C,

(iii) calculate the coordinates [p(x)]C∈R3 of p(x)∈P2 with respect to C.

Let W⊆R

5

W⊆R5 be the set of solutions of the linear homogeneous system given by

x1−x2+4x3−x4−x5=0

−x1+x2−x3+2x4+x5=0

x3−3x4+x5=0

Accordingly,

(i) show that W⊆R5 is a subspace,

(ii) find a basis for W

(iii) dim(W)=?

find the minimal polynomial of the linear operator t : R³ "- R³" define by t (x,y,z) =(x+2y+3z, 4y+5z,6 z).is t

find the minimal polynomial of the linear operator t : r³ "-r³" define by t (x,y,z) =(x+2y+3z, 4y+5z,6 z).is t

Suppose A is an n×n matrix, and let v1,.....vn belong to R^n. Suppose {Av1,.....Avn} is linearly independent prove that A is non singular

Suppose U and V are subspace of R^n with U intersection V={0}. if {u1,.....uk} is a basis for U and {v1,.....vL} is a basis for V, prove that {u1.....uk,v1.......vL} is a basis for u+v.

Find eigen values and associated eigen vectors of the matrix

A=[ 1 −3 3

−3 −5 3

6 −6 4 ] in the field R. Also find an invertible matrix p such that p−1 Ap is diagonal.

Determine wheather the following sets are subspaces of R3

{(a,b,c) : a2+ b2+ c2 ≤1,a,b,c ∈R}

Suppose U and V are subspace of R^n. Prove that orthogonal of ( U intersection V)=orthogonal of U+ orthogonal of V