Elementary Linear Algebra: Applications Version
12th Edition
ISBN: 9781119282365
Author: Howard Anton, Chris Rorres, Anton Kaul
Publisher: WILEY
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Let V be the set of all ordered pairs of real numbers, and consider the following addition and
scalar multiplication operations on u = (u1, u2) and v = (v1, v2) :
u +v= (1+ v1 – 1, u2 + V2 – 1), ku = (ku1, ku2)
(a) Compute u + v and ku for u = (1,2), v = ( – 1,3) and k = 2.
Solution:
%3D
%3D
u + v=
ku
(b) Show that (0, 0) 0.
Solution : We have (u1, u2) + (0,0) = (
); thus u + (0,0) u for every
||
u in V.
) = 0.
(c) Show that (
Solution: We have (u1, u2) + (
thus u + (
(u1
) = (u1, u2);
U2
= u for every u in V.
%3D
u such that u + (- u) = 0
(d) Show that Axiom 5 holds by producing an ordered pair
for u = (u1, u2).
Solution: If u = (u1, u2), then
satisfies u + (- u) = (
-
6.
) = 0;
thus Axiom 5 holds.
(e) Find two vector space axioms that fail to hold.
Solution: Axioms 7 fail to hold. For example:
k(u + v) = k( a
) (
= (ku1, ku2) + (kv1, kv2) = (
%3D
ku +k v
||
6.
k(u + v) ku +kv, k 1.
Axioms 8 fail to hold. For example:
= (k+1)(u1, u2) = (
ku + lu = (kuj, ku2) + (lu1,…
Let V be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication
operations on u=(u1, u2) and v = (v1, V2):
u+v=(u1 +vi + 1, u2 + v2 + 1)
ku = (ku,, ku2)
Compute ku + kv.
(ku1 + kv1, ku2 + kv2)
(ku2 + kv2, ku1 + kv1)
(ku1 + kv1 + 1, ku2 + kv2 + 1)
(ku2 + kv2 + 1, ku1 + kv1 + 1)
If V is not a vector space, which property fails to hold?
k(u + v) = ku + kv
k(mu) = (km)u, where k and m are scalars
%3D
1u = u
none of the above
Is Va vector space with the stated operations?
Yes
No
Let V be the set of all ordered pairs of real numbers with addition and scalar multiplication
and v=
and v= (v₁,v₂) inv
operations defined as follows on V: for u= (₁.₂)
=(₁,4₁₂)
u + v = (u ₁₁ u ₂ ) + ( v₁, v ₂) = (u ₁+ v₁ + 2,u₂+ v₂ − ¹)
and
ku=k(u,,u₂) = (ku,+2k-2,ku ₂-k+1).
This forms a vector space.
Complete the following:
2(1,-1)+(2,3) = (a,b)
The zero vector,0= (c,d)
3=
b=
CF
d=
s
Chapter 4 Solutions
Elementary Linear Algebra: Applications Version
Ch. 4.1 - Let V be the set of all ordered pairs of real...Ch. 4.1 - Let V be the set of all ordered pairs of real...Ch. 4.1 - In Exercises 312, determine whether each set...Ch. 4.1 - In Exercises 312, determine whether each set...Ch. 4.1 - In Exercises 312, determine whether each set...Ch. 4.1 - In Exercises 312, determine whether each set...Ch. 4.1 - In Exercises 312, determine whether each set...Ch. 4.1 - In Exercises 312, determine whether each set...Ch. 4.1 - Verify Axioms 3, 7, 8, and 9 for the vector space...Ch. 4.2 - In Exercises 12, use the Subspace Test to...
Ch. 4.2 - In Exercises 12, use the Subspace Test to...Ch. 4.2 - In Exercises 34, use the Subspace Test to...Ch. 4.2 - In Exercises 56, use the Subspace Test to...Ch. 4.2 - In Exercises 78, use the Subspace Test to...Ch. 4.2 - In Exercises 78, use the Subspace Test to...Ch. 4.2 - In Exercises 1112, use the Subspace Test to...Ch. 4.2 - In Exercises 1112, use the Subspace Test to...Ch. 4.2 - In Exercises 1314, use the Subspace Test to...Ch. 4.2 - In Exercises 1516, use the Subspace Test to...Ch. 4.2 - In parts (a)(h) determine whether the statement is...Ch. 4.3 - Which of the following are linear combinations of...Ch. 4.3 - Express the following as linear combinations of u...Ch. 4.3 - Prove that if {u, v} spans the vector space V,...Ch. 4.4 - Explain why the following form linearly dependent...Ch. 4.4 - In each part, determine whether the vectors are...Ch. 4.4 - In each part, determine whether the vectors are...Ch. 4.4 - In each part, determine whether the vectors are...Ch. 4.4 - In each part, determine whether the matrices are...Ch. 4.4 - a. Show that the three vectors v1 = (0, 3, 1, 1),...Ch. 4.4 - Prove that if {v1, v2} is linearly independent and...Ch. 4.5 - Use the method of Example 3 to show that the...Ch. 4.5 - Use the method of Example 3 to show that the...Ch. 4.5 - Show that the following polynomials form a basis...Ch. 4.5 - Prob. 5ECh. 4.5 - Show that the following matrices form a basis for...Ch. 4.5 - In each part, show that the set of vectors is not...Ch. 4.5 - Show that the following vectors do not form a...Ch. 4.5 - Show that the following matrices do not form a...Ch. 4.5 - Find the coordinate vector of w relative to the...Ch. 4.5 - Find the coordinate vector of v relative to the...Ch. 4.5 - Find the coordinate vector of p relative to the...Ch. 4.5 - In Exercises 1516, first show that the set S =...Ch. 4.5 - In Exercises 1718, first show that the set S =...Ch. 4.5 - In words, explain why the sets of vectors in parts...Ch. 4.5 - In any vector space a set that contains the zero...Ch. 4.5 - The accompanying figure shows a rectangular...Ch. 4.5 - The accompanying figure shows a rectangular...Ch. 4.5 - Consider the coordinate vectors...Ch. 4.5 - The basis that we gave for M22 in Example 4...Ch. 4.5 - Prove that if V is a subspace of a vector space W...Ch. 4.5 - In parts (a)(e) determine whether the statement is...Ch. 4.5 - In parts (a)(e) determine whether the statement is...Ch. 4.5 - In parts (a)(e) determine whether the statement is...Ch. 4.5 - In parts (a)(e) determine whether the statement is...Ch. 4.5 - In parts (a)(e) determine whether the statement is...Ch. 4.6 - In Exercises 16, find a basis for the solution...Ch. 4.6 - In Exercises 16, find a basis for the solution...Ch. 4.6 - In Exercises 16, find a basis for the solution...Ch. 4.6 - In each part, find a basis for the given subspace...Ch. 4.6 - Find the dimension of each of the following vector...Ch. 4.6 - a. Show that the set W of all polynomials in P2...Ch. 4.6 - Find a standard basis vector for R3 that can be...Ch. 4.6 - Let {v1, v2, v3} be a basis for a vector space V....Ch. 4.6 - The vectors v1 = (1, 0, 0, 0) and v2 = (1, 1, 0,...Ch. 4.6 - Find a basis for the subspace of R4 that is...Ch. 4.6 - Let S be a basis for an n-dimensional vector space...Ch. 4.6 - Let S = {v1, v2,, vr} be a nonempty set of vectors...Ch. 4.6 - State the two parts of Theorem 4.6.2 in...Ch. 4.6 - In each part, let S be the standard basis for P2....Ch. 4.6 - In parts (a)(k) determine whether the statement is...Ch. 4.6 - In parts (a)(k) determine whether the statement is...Ch. 4.6 - In parts (a)(k) determine whether the statement is...Ch. 4.6 - In parts (a)(k) determine whether the statement is...Ch. 4.6 - In parts (a)(k) determine whether the statement is...Ch. 4.8 - In Exercises 34, determine whether b is in the...Ch. 4.8 - In Exercises 78, find the vector form of the...Ch. 4.8 - In Exercises 910, find bases for the null space...Ch. 4.8 - In Exercises 1112, a matrix in row echelon form is...Ch. 4.8 - In Exercises 1415, find a basis for the subspace...
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