linear-guest

2.6 Review Problems 69
so that a 2 jx j ≡ a 2 i x i ; this is called “relabeling dummy indices”. When
dealing with products of sums, you must remember to introduce a
new dummy for each term; i.e., a i x i b i y i = ∑ i a ix i b i y i does not equal
a i x i b j y j = ( ∑ i a ix i)( ∑ j b jy j) .
Use Einstein summation notation to propose a rule for Mx so that
Mx = 0 is equivalent to the linear system
a 1 1x 1 +a 1 2x 2 · · · +a 1 k xk = 0
a 2 1x 1 +a 2 2x 2 · · · +a 2 k xk = 0
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a r 1x 1 +a r 2x 2 · · · +a r k xk = 0
Show that your rule for multiplying a matrix by a vector obeys the
linearity property.
4. The standard basis vector e i is a column vector with a one in the ith
row, and zeroes everywhere else. Using the rule for multiplying a matrix
times a vector in problem 3, find a simple rule for multiplying Me i ,
where M is the general matrix defined there.
5. If A is a non-linear operator, can the solutions to Ax = b still be written
as “general solution=particular solution + homogeneous solutions”?
Provide examples.
6. Find a system of equations whose solution set is the walls of a 1 × 1 × 1
cube. (Hint: You may need to restrict the ranges of the variables; could
your equations be linear?)
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