Consider the homogeneous linear equation Ax = 0, (when the right-hand-side vector is the zero vector, the equation is called homogeneous.) If A has linearly independent columns, then the solution of Ax = 0 can only be the zero vector. Can you see why? It is because Ax = [a₁, a2,, an] 21 x₁α₁ + + Cnn is essentially a linear combination of the columns of A, where the scaling coefficient of the i-th columna is exactly S Xn the i-th component x; in the solution. When the LC sums up to the RHS zero vector 0, the linear independence of the columns of a1, a2,, an would force the coefficients #₁,...,n to be all zeros! (Otherwise it means the columns of A are linearly dependent.) Now try the above theoretical reasoning to the following equation with a concrete A 1 0 0 1 1 0 1 1 1 1 0 1 Ax = x1 x2 0 Pay attention to the dimension matches here (explicitly written out for you already). Multiply Ax out using standard matrix-vector product, then use LC of column to express Ax, you should get the same product! Then make it equal to the RHS zero vector, then find the solution: 1 type your answer... 2 type your answer... 3 type your answer... Spend some time to convince yourself that this is the only solution you can get. Can any other solution for the above homogeneous linear equation Ax = 0 exist? type your answer... (input only yes or no)

Find the general solution of the simple homogeneous "system" below, which consists of a single linear equation. Give your answer as a linear combination of vectors. Let x2 and x3 be free variables. 4x1-16x₂ + 12x3 = 0 X₁ X2 =x₂ “EHOHO X3 (with X2, X3 free) O A. O C. X₁ X2 =X₂ 4 0 1 X3 (with X2, X3 free) + X3 - 3 0 C X₂ THI X2 1 = x₂ +X3 0 0 X3 (with X₂, X3 free) O B. O D. X₁ X2 X₁ = 4 X2 +3 X₂ X3 X3 X3 (with X2, X3 free) -3 X₁ 1

Determine if the following sets of vectors are linearly dependent or linearly independent. Why or why not? (Be sure to reduce each corresponding matrix to echelon form as part of your justification.) (a) (b) 1 5 000 4 1 (c) 3 7 0 (QQ-B) 3 -2 -3 9 1 -008) 3 -2 4 1

Solve the set of equations to vector form (a b c d)+lambda(e f g h)