Linear Algebra Answers

1. the square root of (2-i) is?

2. for a given ,matrix A=[(5,-6),(-3,2)] the matrix P that diagonalizes A is?

3. for a,b is an element of R, let S be a subset of R^2 defined as S={(x,y) is an element of R^3:x+y+axy=b}. Then S is a subspace of R^2 if...?

4. suppose U={(x,y,x+y,z,2y+z) is an element of F^5:x,y,z is an element of F}, then a subspace W of F^5 such that F^5=U denote W is...?

5. the vectors (1,-1,2),(2,3,1),(3,2,t) are not basis of R^3 if...?

1. in R3, let U span (1,0,0),(0,1/root2, 1/root2). then U is an element of U such that ||u-(2,4,6)|| is as small as possible.
2. for a given function f:R to R defined as f(x)=2x-1, the image of S={x is an element of R: x^2-4>/=0} is?
3. suppose T: R^2 to M22 is a linear defined by T(U,V)=[(U,U), (V,2U)]. Then ker(T) is?
4. suppose T:R^6 to R^4 is a linear map such that null T=U, where U is 2-dimensional subspace of R^6. Then dim range T is?

LA. 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

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.