Linear Algebra Answers

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

Can you construct a linear transformation T : R (3) 2 → R

3

such that

Im(T) = {(x, y,z) ∈ R

3

: ax + by + cz = 0} where a, b, c ∈ R are constants?

Solve the simultaneous equation +=5 and +=1

Suppose u, v ∈ V and ||u|| = ||v|| = 1 with < u, v > = 1.

Prove that u = v.

Please assist.

1.find an expression for 1/2||u+v||²+1/2||u-v||² in terms of ||u||²+||v||².

2.find an expression for ||u+v||²-||u-v||² in terms of u×v.

3.use the result of 2 to deduce an expression for ||u+v||² whenever u and v are orthogonal to each other.

Suppose S,T element of L(V) are such that ST = TS. Prove that null S is invariant under T.

Prove that there does not exist a linear map T : R5 - R5 such that range T = null T.

Suppose b,c element of R, and T: R³ - R² dened as T (x;y;z) = (2x4y+3z+b,6x+cxy): Show that T is linear if and only if b = c = 0.

Suppose V is finite-dimensional with dim V ≥ 2.

Prove that there exist S, T ∈ L(V; V ) such that ST ≠ T S.

please assist.