Linear Algebra Answers

Let T€L(R) such that - 4,5 and root 7 are its eigenvalues. Show that T(x) - 9x=(- 4,5, root 7)

(4.1) Consider the point A = (−1,0,1), B = (0,−2,3), and C = (−4,4,1) to be vertices of a triangle ∆. Evaluate all side lengths of ∆. Let ∆ be the triangle with vertices the points P = (3,1,−1), Q = (2,0,3) and R = (1,1,1). (4.2)Determine whether ∆ is a right angle triangle. If it is not, explain with reason, why?

(4.3)Let u =< 0,1,1 >,v =< 2,2,0 >and w =< −1,1,0 >bethreevectorsin standard form.

(a) Determine which two vectors form a right angle triangle?

(b)Find θ := uw, the angel between the given two vectors.

(4.4) Let x < 0. Find the vector n =< x,y,z > that is orthogonal to all three vectors u =<1,1,−2 >,v =< −1,2,0 >and w =< −1,0,1 >.

(4.5) Find a unit vector that is orthogonal to both u =< 0,−1,−1 > and v =< 1,0,−1 >

1.let x<0.find the vector n=<x,y,z> that is orthogonal to all three vectors u=<1,1,-2>,v=<-1,2,0> and w=<-1,0,1>.

2.find a unit vector that is orthogonal to both u=<0,-1,-1> and v=<1,0,-1>.

1.let u=<0,1,1>,v=<2,2,0> and w=<-1,1,0> be three vectors in standard form.

1.1.determine which two vectors form a right angle triangle?

1.2.find @:u w, the angel between the given two vectors.

Suppose V is finite-dimensional with dim V greater or equal to 2. Prove that there exist S, T €L(V, V) such that ST is not equal to T S

Suppose V is finite-dimensional with dim V greater or equal to 2. Prove that there exist S, T €L(V, V) such that ST is not equal to T S

Verify rank nullity theorem for the linear 

transformation T : R

3 → R

3

 defined by : 9 

T(x, y, z) = (x + 2y – z, y + z, x + y – 2z)

Evaluate 𝒂 so that the sum of the eigen values of 𝑨 is 10. [ 𝑎 4 −2 1 3 0 −6 4 𝑎 ]

a) Determine whether or not the following are subspaces?

i. 𝑾 = {(𝒂,𝒃, 𝒄) ∈ ℝ𝟑

|𝒂 + 𝒃 + 𝒄 = 𝟎} of ℝ𝟑

ii. The symmetric matrices of 𝑴𝒏𝒏 (the vector space of 𝒏 × 𝒏

matrices)

iii. All polynomials of degree 2.

b) 

i. For which real values of 𝝀 do the following vectors form a 

linearly dependent set in ℝ𝟑

?

𝒗𝟏 = (𝝀, −

𝟏

𝟐

, −

𝟏

𝟐

) , 𝒗𝟐 = (−

𝟏

𝟐

, 𝝀, −

𝟏

𝟐

) , 𝒗𝟑 = (−

𝟏

𝟐

, −

𝟏

𝟐

, 𝝀)

ii. Find a basis and dimension of the solution space for the 

following homogenous linear equations: 

𝒙𝟏 + 𝟐𝒙𝟐 − 𝒙𝟑 + 𝟒𝒙𝟒 = 𝟎

𝟐𝒙𝟏 − 𝒙𝟐 + 𝟑𝒙𝟑 + 𝟑𝒙𝟒 = 𝟎

𝟒𝒙𝟏 + 𝒙𝟐 + 𝟑𝒙𝟑 + 𝟗𝒙𝟒 = 𝟎

𝒙𝟐 − 𝒙𝟑 + 𝒙𝟒 = 𝟎

𝟐𝒙𝟏 + 𝟑𝒙𝟐 − 𝒙𝟑 + 𝟕𝒙𝟒 = 𝟎