1. Span: intuition and calculations. Consider the two vectors as an (a) Write Span{a1, a2} as a linear combination of a1 and 212, using scalar variables 2:1 and :32 to denote the weights. (b) Write the linear combination of a1 and a2 in the form of matrixvector multiplication, as Ax, . $1 us1ng the vector x : l ] . 372 (c) Do these vectors reside in R, or R9, or R3, or another R\"? How do you know? (d) Sketch the vectors a1 and a2, and Span{a1, 3.2}, on a 2D plot. (e) Is Span{a1, a2} a line, or a plane, or a volume? Why? Now consider this additional vector, where p is unspecied: .3 = [23] (i) For what value(s) of p will Span{a1, a2, a3} = Span{a1, a2} ? You must show calculations here. (g) Choose another value of p that is different from your answer above and plug in to a3. Now, is Span {a1, a2, a3} a line, or a plane, or a volume? Why? You should not have to perform calculations here, but please provide a justification. Next, choose p = 3 for an example, so that a3 = 3 (h) Sketch the vectors a1, a2, a3 and Span{a1, a2, a3}, on a 2D plot. Note: the book has several good examples of plotting similar spans in Ch. 1.3. (i) Lastly, here is a fourth and final vector: b = ON Using your plots, is this b in Span {a1, a2}, or Span {a1, a2, a3}, or both, or neither? Why? Note: a description in words is fine for this problem, no need to perform calculations