Linear Algebra Answers

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

Determine wheather the following sets are subspaces of R

3

{(a,b,c) : a

2+ b

2+ c

2 ≤1, a,b,c ∈R}

Let U and W be subspaces of a vector space V of finite dimension. Prove that(UnW)°=U°+W°

Find a basis for the following subspace of

R

5

.

J

=

{

⃗

x

∈

R

5

∣

x

1

=

x

2

=

x

5

,

x

3

+

x

4

=

}

What is the dimension of

J

?

Express V= 2t² + 5t + 9 as a linear combination of the polynomials

P1= t + 1

P2= t - 1

P3 = t² - 2t + 1

Check that {1,(x+1),(x+1)^2} is a basis of the vector space of polynomial over R of degree at most 2. Find the coordinate of 3+x+2x^2 with respect to the basis.

Q(1) Determine Whether Each Of The Following Systems Is Linear: (A) 3x – 4y + 2yz = 8 (b) ex + 3y = 1(C) 2x-3y+kz=4

Q(2)  Write V As A Linear Combination Of U1, U2, And U3 V=(1,4,6) U1=(1,1,2) U2=(2,3,5) U3=(3,5,8)