Summary
Highlights
Professor Gilbert Strang introduces MIT's 18.06 Linear Algebra course, discussing the textbook and course web page. He outlines the lecture's plan: to solve a system of linear equations using the Row picture, the Column picture, and the matrix form.
An example system of two linear equations with two unknowns (2x - y = 0 and -x + 2y = 3) is presented. The matrix form AX=B for this system is briefly introduced. The discussion then focuses on the Row picture, where each equation represents a line in the xy-plane. The solution is the intersection point of these two lines, found to be (x=1, y=2).
The column picture is introduced as a more crucial perspective. The system of equations is reinterpreted as finding a linear combination of column vectors that equals the right-hand side vector. The column vectors are drawn in the xy-plane, and it's shown how a specific linear combination (1 times column 1 + 2 times column 2) graphically results in the right-hand side vector (0, 3). The concept of linear combination as the most fundamental operation in linear algebra is emphasized.
The lecture moves to an example of three linear equations with three unknowns (x, y, z). The corresponding 3x3 matrix form AX=B is shown. In the Row picture, each equation now represents a plane in 3D space. The intersection of three such planes ideally yields a single point, which is the solution. However, visualizing the intersection of three planes becomes complex, highlighting the limitations of the Row picture in higher dimensions.
The column picture is applied to the 3x3 system. The equation is seen as finding a linear combination of three 3-dimensional column vectors that equals the right-hand side vector. A specific right-hand side is chosen to illustrate a simple solution: x=0, y=0, z=1, which means the right-hand side is exactly one of the column vectors. This reinforces the idea of linear combinations.
The crucial question is posed: Can AX=B be solved for every possible right-hand side vector 'b'? In the context of the column picture, this translates to asking if the linear combinations of the column vectors fill the entire 3D space. For good matrices (non-singular, invertible), the answer is yes. The lecture explains that if the column vectors lie in the same plane, they cannot fill the entire 3D space, leading to cases where no solution exists for certain 'b' vectors.
The concept is extended to 9 dimensions, acknowledging the difficulty of visualization but emphasizing that the underlying idea of linear combinations remains the same. The lecture concludes by clearly defining matrix-vector multiplication (A times x) as a linear combination of the columns of matrix A, with the components of vector x as the coefficients. An example of a 2x2 matrix-vector multiplication is demonstrated, showcasing both the column combination method and the dot product (row by vector) method.