[Personal Notes] Introduction to Linear Algebra by Gilbert Strang — 2.1
Vectors and Linear Equations
Linear algebra solves a system of linear equations where variables like x and y are only multiplied by numbers. The following is a simple linear system with two equations and two unknowns.
Two lines meet at a point that is a solution. The row picture depicts what it is like to look at a row at a time.
Now with the column picture, we turn numbers into vectors to look at a column at a time. In the column picture, we can see how the columns combine to produce the vector b on the right side. Our solution is the right coefficients for the linear combination of the columns. We can turn the linear combination into a matrix equation Ax = b.
Three Equations in Three Unknowns
When there are the same number of equations and unknowns, we usually have one solution.
The row picture of above equations shows a point where three planes meet. Two planes intersect at a line, and that line intersects with the third plane at a point (0, 0, 3).
With the column picture, it is easier to see the solution because it is obvious that we need two of the third column vector to produce the vector b.
The Matrix Form of the Equation
What does it really mean to multiply A times x? From row point of view, Ax is dot products between each row and x. From column point of view, Ax is a linear combination of the columns. We will concentrate more on the column point of view. Identity matrix is a matrix that does not change the vector it multiplies to. Therefore, Ix = x.
Matrix Notation
Each element of matrix A is written as A(i, j) or aij, where i is the row and j is the column.
