MATH08013: In each case determine whether the vectors are linearly dependent or linearly independent: Maths Assignment, UCC, Ireland

Question 8
i. If 𝐴⃗ = 4𝑖⃗ + 𝑗⃗ − 7𝑘⃗⃗ and 𝐵⃗⃗ = 3𝑖⃗ + 1 2 𝑗⃗ − 7𝑘⃗⃗ then find 𝐴⃗ ⋅ 𝐵⃗⃗ , 𝐴⃗ ⋅ 𝐴⃗ , 𝐵⃗⃗ ⋅ 𝐴⃗ , 𝐵⃗⃗ ⋅ 𝐵⃗⃗ , 𝐴⃗ × 𝐵⃗⃗ , 𝐴⃗ × 𝐴⃗ , 𝐵⃗⃗ × 𝐴⃗ , and 𝐵⃗⃗ × 𝐵⃗⃗ .

Hint: use the commands of dot(A, B) and cross(A, B) in MATLAB

ii. In each case determine whether the vectors are linearly dependent or linearly independent:
(a) 𝐴⃗ = 2𝑖⃗ + 𝑗⃗ − 3𝑘⃗⃗ , 𝐵⃗⃗ = 𝑖⃗ − 4𝑘⃗⃗ , 𝐶⃗ = 4𝑖⃗ + 3𝑗⃗ − 𝑘⃗⃗
(b) 𝐴⃗ = 𝑖⃗ − 3𝑗⃗ + 2𝑘⃗⃗ , 𝐵⃗⃗ = 2𝑖⃗ − 4𝑗⃗ − 𝑘⃗⃗ , 𝐶⃗ = 3𝑖⃗ + 2𝑗⃗ − 𝑘⃗⃗

Hint: use the command of rref(matrix) and cross(A, B) in MATLAB. Reduced Row Echelon Form (RREF) takes the Gauss–Jordan elimination method one step further by performing scaling EROs on all rows so that all coefficients on the diagonal become one.