Linear Algebra Answers

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)

Show that transformation 𝑇:𝑅

2 → 𝑅

2 defined by 

𝑇(𝑥, 𝑦) = (𝑎𝑥 + 𝑏𝑦, 𝑐𝑥 + 𝑑𝑦),

If the matrix R =[-5 2]

[M -4]

is singular, determine the

value of m .

Are there values of r and s for which [1,0,0,0,r-2,2,0,s-1,r+2,0,0,3] has rank 1 or 2? If so, find those values