Linear Algebra Answers

Obtain a solution set for the linear system

x1-2x2-3x3=0

-2x1+4x2+6x3=0

X1+2x2-5=0

Let T and U be the linear operator on R^2 defined by

T(x1,x2)=(x2,x1) and U(x1,X2)=(x1,0)

A) How would you describe T and U in geometrically

B) give rules like the ones defining..

Does the basis 

B = {(1, 0, 1), (1, 0, — 1), (0, 3, 4)} 

form an orthonormal basis of R3 with 

respect to the standard inner product of 

R3 ? Justify your answer. If it doesn't form 

an orthonormal basis for R3, apply 

Gram-Schmidt process to obtain an 

orthonormal basis R3 with respect to the 

standard inner product on R3. 

Let A and B be n × n matrices with A invertible. Prove that AB and BA have the

same eigenvalues. 

Let V be a finite dimensional inner product space over the field C of complex

numbers. Suppose B = {x1, x2, . . . , xn} is an orthonormal basis of V .

Prove that for all x, y ∈ V ,

(x|y) = Xn

i=1

(x|xi)(y|xi).

Let V be the vector space of real 2 × 2 matrices with inner product

(A|B) = tr(B

tA).

Let U be the subspace of V consisting of the symmetric matrices. Find an orthog-

onal basis for U

⊥ where U

⊥ = {A ∈ V | (A|B) = 0 ∀B ∈ U}.

Find all solutions to the following system of linear equations.

x1 − x2 − x3 = 2

2x1 + x2 + 2x3 = 4

x1 − 4x2 − 5x3 = 2

𝑇: ℝ3→ℝ3 defined by 𝑇(𝑥1,𝑥2,𝑥3)=(𝑥1+𝑥2,𝑥2+𝑥3,𝑥3+𝑥1).

Let W be a subspace of R^5, which is spanned by the vectors u₁ = (1, 2, 1, 0, 0) u₂=(0, 1, 3, 3, 1) u₃= (1, 4, 6, 4, 1)

Find a basis for W₀

Find the basis (a_1, a_2, a₃) that is dual to the following basis of R³

{u₁ = (1, -1, 3), u₂ = (0, 1, -1), u₃ = (0, 3, -2)}