Lecture 1: 1.1-1.2: Solving systems of linear equations, Complex numbers

Prove: If A is k×n matrix, x∈Rn, b∈Rk A*x=b is a system of k lin equations ofn variables

Def: A*x=(x_1)*(a_1)+(x_2)*(a_2)+…+(x_n)*(a_n)=b
A:((a_1),(a_2),…(a_n))

Ex:

{[2*(x_1)−(x_2)+(x_3)=(b_1)],[(x_1)+(x_2)−(x_3)=(b_2)],[2*(x_2)−(x_3)=(b_3)])*A=([2,−1,1],[1,1,−1],[0,2,−1])*([(x_1)],[(x_2)],[(x_3)])=([5],[9],[0])

A=[[2,-1,1],[1,1,-1],[0,2,-1]]→row reduce (change (R_1)and ((R_2)) [[1,1,-1],[2,-1,1],[0,2,-1]]→(R_2)-2*(R_1)*[[1,1,-1],[0,-3,3],[0,2,-1]]→-1/3*(R_2)*[[1,1,-1],[0,1,-1],[0,2,-1]]

→(R_3)-2*(R_2)*[[1,1,-1],[0,1,-1],[0,0,1]] so A*x = b consistent for any b

Ex:
A=[[1,1,3],[-1,0,1],[0,1,4]]--> (R_2)+(R_1) [[1,1,3],[0,1,4],[0,1,4]]--> (R_3)- (R_2)*[[1,1,3],[0,1,4],[0,0,0]]

2 pivots for some b
A*x=b inconsistent

Ex:

([2,1],[0,1],[−1,1])*([1],[2])=1*([2],[0],[−1])+2*([1],[1],[1])=([2],[0],[−1])+([2],[2],[2])=([4],[2],[1])

([-2,1],[0,1],[−1,1])*([−1],[−2])=−1*([−2],[0],[−1])+(−2)*([1],[1],[1])=([−4],[−2],[1])

1.5 Linearity of matrix multiplication

Let A be a matrix and x,y∈Rn.

1. Scalar multiplication

A*(c*x)=c*(A*x)

Proof:

Let

x=([(x_1)],[(x_2)],[⋮],[(x_n)])

Then

A*(c*x)=A*([c*(x_1)],[c*(x_2)],[⋮],[c*(x_n)])=c*(x_1)*(a_1)+c*(x_2)*(a_2)+⋯+c*(x_n)*(a_n)

Factor out c:

=c*((x_1)*(a_1)+(x_2)*(a_2)+⋯+(x_n)*(a_n))=c*(A*x)

2. Additivity

A*(x+y)=A*x+A*y

Proof: