Page 69 - 35Linear Algebra

2.6 Review Problems 69

2 j
2 i
so that a x ≡ a x ; this is called “relabeling dummy indices”. When
j i
dealing with products of sums, you must remember to introduce a
i
i
i
i
new dummy for each term; i.e., a i x b i y = P a i x b i y does not equal
i
j
i
P
P
j
a i x b j y = a i x i b j y .
i j
Use Einstein summation notation to propose a rule for Mx so that
Mx = 0 is equivalent to the linear system
1 2
1 k
1 1
a x +a x · · · +a x = 0
1
2
k
2 1
2 2
2 k
a x +a x · · · +a x = 0
2
1
k
. . . . . . . . . . . .
r k
r 2
r 1
a x +a x · · · +a x = 0
2
k
1
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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