Linear Algebra Answers

Solve x y z and t in the matrix equation below [3x t+

There are at least three different unitary matrices of order 2. True or false with full explanation

Find the vector equation of the plane determined by the points (-1,-2,1),(1,0,1) and (1,-1,1). Also check whether (1/2,1/2,1/2) lies on it

Suppose A, B, C and D are matrices with the following sizes A(5x2), B(4x2), C(4X5), and D(4x5)

Determine each of the following case whether a product is defined. If it is so then give the size of the resulting matrix

a. DC

b. -CA+B

c. CD-D

Suppose U₁; U₂;...; U_m are finite-dimensional subspaces of V .

Prove that U₁ + U₂ +...+ U_m is finite-dimensional and dim(U₁ + U₂ +.....+ U_m) ≤ dim U₁ + dim U₂ +...+ dim U_m.

Let U be the subspace of R⁵ denoted by: U = { (x₁; x₂; x₃; x₄; x₅) ∈ R⁵ : x₁ = 3x₂ and x₃ = 7x₄

1)   Find a subspace W of R⁵ such that R⁵ = U ⊕ W.

Determine whether the function T: R

3 →R

3 by 𝑇 ([

𝑥

𝑦

𝑧

]) = [

𝑥 + 𝑦 − 𝑧

2𝑥𝑦

𝑥 + 𝑧 + 1

] is a linear 

transformation between vector space.

Using gaussian elimination method find all solutions to the following system of linear equations.

2x₂ + 3x₃ + 4x₄ = 1

x₁ - 3x₂ + 4x₃ + 5x₄ = 2

-3x₁ + 10x₂ - 6x₃ -7x₄ = -4

Let A = [ 0 1 2

1 0 3

4 -3 8 ]

If A is an invertible matrix, then show that.

det (A-1) = 1/det ( A )

Find the solution of the following system of equation by substitution method.

2x + 5y + 2z = -38

3x - 2y + 4z = 17

-6x + y - 7z = -12