Lec 1 | MIT 18.06 Linear Algebra, Spring 2005
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Further reading 5
Can changing representation reveal a solution? →A different representation makes certain relationships directly visible. Compare which calculation each representation makes easier and which information it preserves.How can columns explain a matrix equation? →Translate coordinates into a geometric picture. The columns of a matrix then show which combinations of vectors can reach the desired output.Which equations survive a collapse in dimension? →Once a transformation flattens space, reachable outputs lie in its image. A target inside that image and one outside it lead to different answers.What is the same in a diagram and a column calculation? →Switch between geometric motion and coordinate arithmetic to see one linear operation from two angles. Strang’s column combination is a worked example; neither representation is best for every task.Which outputs disappear when a transformation collapses space? →Some target vectors fall outside the image after a collapse. In the eigenvalue example it is A minus lambda times the identity that becomes singular; this does not make every system unsolvable.
Caption notes
Read Ax=b as intersections of constraints or as combinations of matrix columns. Dependent columns leave some targets unreachable.
Original passages · 5
A times x is a combination
of the columns of A.If these three columns
all lie in the same plane, then their combinations
will lie in that same plane.this row picture is getting a little hard to see.the most fundamental operation in the whole courselinear combinations of the columns fill three dimensional space