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numbers, giving dollar values in millions, are excerpted from [Leontief 1965],
describing the 1958 U.S...
(b) Solve for total output if next year's external demands are: steel's demand
up 10% and auto's demand up 15%...
Linear Systems
experts is a variation on the code above that first finds the best pivot among
the candidates, and then scales it to a number that is less likely to give trouble...
For each electrical component there is a constant of
proportionality, called its resistance, satisfying that potential = flow-resistance...
(d) What is the formula for more than two resistors in parallel?
6 In the car dashboard example that begins the discussion, solve for these amper-
ages...
The step
up in abstraction from studying a single space at a time to studying a class
of spaces can be hard to make...
Prove that this is not a vector space: the set of two-tall column vectors with
real entries subject to these operations...
(b) What if it doesn't contain the origin?
/ 1.38 Using the idea of a vector space we can easily reprove that the solution set of
a homogeneous linear system has either one element or infinitely many elements...
The
vector space of cubic polynomials {a + bx + ex2 + dx3 a, b,c,d <G R} has a sub-
space comprised of all linear polynomials {m + nx m,n el}...
The fifth condition is
satisfied because for any s <G S, closure under linear combinations shows that
the vector 0 • 0 + (—1) • s is in S; showing that it is the additive inverse of s
under the inherited operations is routine...
Now the subspace is described as the collection of unrestricted linear combi-
nations of those two vectors...
Does every non-
trivial space have infinitely many subspaces?
2.32 Finish the proof of Lemma 2.9...
(a) If S C T are subsets of a vector space, is [S] C [T]? Always? Sometimes?
Never?
(b) If S, T are subsets of a vector space, is [S U T] = [5] U [T]?
(c) If S, T are subsets of a vector space, is [S D T] = [S] n [T]?
(d) Is the span of the complement equal to the complement of the span?
2.46 Reprove Lemma 2.15 without doing the empty set separately...
Lemma A subset S of a vector space is linearly independent if and only if
for any distinct 5*1,..., sn E S the only linear relationship among those vectors
C1S1 H h CnSn =0 Ci,...,CnGl
is the trivial one: c\ = 0,..., cn = 0...
Similarly, working straight from the definition, a set with four vectors
would require checking three vector equations...
This last set is
linearly independent (this is easily checked), and so removal of any of its vectors
will shrink the span...
Applying Lemma 1.1, we conclude that if
S is linearly independent and v &" S then S U {v} is also linearly independent
if and only if [S U {v}] ^ [S]...
Vector Spaces
(a) {2,4sin2(cc),cos2(cc)} (b) {1, sin(cc), sinBcc)} (c) {x, cos(cc)}
(d) {(l + xJ,x2 + 2cc,3} (e) {cosBx),sin2(V),cos2(V)} (f) {0,x,x2}
1.22 Does the equation sin2(cc)/cos2(cc) = tan2(cc) show that this set of functions
{sin2(cc), cos2(cc), tan2(cc)} is a linearly dependent subset of the set of all real-valued
functions with domain (—7r/2..7r/2)?
1.23 Why does Lemma 1.4 say "distinct"?
•/ 1.24 Show that the nonzero rows of an echelon form matrix form a linearly inde-
pendent set...
(The requirement
that a basis be ordered will be needed, for instance, in Definition 1.13.)
1.2 Example This is a basis for R2...
The down side is that representations look like vectors from M.n7 and that
can be confusing when the vector space we are working with is W1, especially
since we sometimes omit the subscript base...
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