Linear algebra 1. The given set with the given operations is not a vector space. Check all the properties and list whichproperties fail to hold.The set of all ordered pairs of real numbers with the operations (x,, y,) O (x,, y2) = (x, +x,,y, + y,) andrO(x, y) = (0,0).2. Prove that a vector u in a vector space has only one negative -u. [Hint: Assume that there are twonegatives and conclude that they must be equal.]3. C(-00, 00) denotes the vector space of real-valued continuous functions.Which of the following subsets are subspaces of the vector space C(-x,0) ? Explain.a) All nonnegative functions.b) All constant functions.c) All functions f such that f(0) = 0.d) All functions f such that f(0) = 5.e) All differentiable functions.4. a) Prove that proj (proj-v) = proj,vb) Prove that proj (v – proj v) =0.c) Explain parts a) and b) geometrically.5. a) Prove that u-v2u + v for all vectors in R".b) Explain parts a) geometrically.

Answered: Image /qna-images/answer/4e1791e1-bdee-49bc-8c00-0171cf1aa0c3.jpg

1. The given set with the given operations is not a vector space. Check all the properties and list which properties fail to hold. The set of all ordered pairs of real numbers with the operations (x,, y) (x,,y;) = (x, +x,, y, +y,) and rO(x, y) = (0,0).

1. The given set with the given operations is not a vector space. Check all the properties and list which properties fail to hold. The set of all ordered pairs of real numbers with the operations (x,, y) (x,,y;) = (x, +x,, y, +y,) and rO(x, y) = (0,0).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter8: Applications Of Trigonometry

Section8.4: The Dot Product

Problem 48E

See similar textbooks

Related questions

Q: Let v, = V2 = - 2 and w = - 1 Determine if w is in the subspace of R° generated by v, and v2. 3 - 5…

Q: Let W be the subspace spanned by the given vectors. Find a basis for W. 1 -1 1 %3D ,W2 = %3D 4 -3 -2…

A: The given vectors are: w1=1-14-2 and w2=01-31

Q: 10 -7 Let u = and v = Compute u • v, v · v, ||u|| and find the dis- 13 16 tance between u and v.

A: Given u=10-7135 and v=-44316

Q: Find a basis for the eigenspace corresponding to each listed eigenvalue of A below. 3 0 A = 1 = 2,3…

A: Eigen Space: For a matrix A and an eigenvalue λ∈F(Field), Eλ=x∈V Ax=λx , where V is a vector space,…

Q: (b) Find the characteristic equation of the matrix P defined by [2 1 11 P = 1 2 1 li 1 21 Further…

A: The given matrix is P=211121112 the characteristic equation is given by P-λI=0 where I is the…

Q: For each problem find square matrices which satisfy the given conditions. You don't have to justify…

A: Since you have posted a question with multiple sub-parts, I will solve the first three subparts for…

Q: Determine the dot products of the following pairs of vectors. (a) (2, 4), (3, -4) (b) (3, –2, 5),…

Q: Consider the vectors V1, V2, V3 and b. 1 2 8 -2 V1 ,V3 -13 6. 12 18 Write b as a linear combination…

Q: Question SHOW THAT THE SET OF TYPE MATRICES "mxn" IS A VECTOR SPACE OF MATRICES TYPE "mxn" (M(mxn)…

Q: Let A be a 3x3 matrix such that Col A = Span and Nul A = Span then Select one: O a. The equation Ax…

A: Linear algebra. Rank nullity theorem.

Q: Define the linear transformations T:R R2 and S: R R so that 3. X1 + X2 -X2 and S X2 -I - 12 a.) Is…

Q: Let 0 3 0 1 1 2 2 1 A = -1 B : 0 1 1 2 0 1 -2 Compute det(AB), det(A-'B-!) and det(2A) where 2A…

Q: Find the kernel and nullity of the transformation T. T(f(t)) = f'"(t) + 4f'(t) from P, to P2 %3D

A: Let T be a linear transformation from a vector space V then according to dimension theorem ,…

Q: linear algebra

A: 6. The given matrix is

Q: Use Cramer's Rule to solve for x2. xi + 2x2 +x3 = 2 3x1 – x2 - 2x3 = 7 X2 +x3 = -1

A: Here we have given one matrix and we have to find the value of x2 by using the Cremer’s rule. Now…

Q: 40 0 -1 2 0 1 2 4 Given A = (a) Find the eigenvalues of A. (b) For each distinct eigenvalue, find…

A: The given problem is to find the eigenvalues and eigenvectors of the given matrix and check whether…

Q: Consider the following bases of R2: {0 A} a = and (a) Find the change of basis matrix Q from a to y.…

Q: 9. Write the complete solution x = x, +x¼ to the given linear system. x + 3y + 3z = 1 2x + 7y + 9z =…

A: Given : Given system of linear equation is x+3y+3z=12x+7y+9z=5-x-3y-3z=-1 We want to write complete…

Q: Find the determinant of the linear transformation. T(f) = 5f – 5f' from P2 to P2

A: Given below the detailed solution

Q: 4. Use Cramer's Rule to find x for the linear system given by x+ 2y – z = -1 3x – y + z = -6 2x - 2z…

A: The given linear system is x+2y-z=-13x-y+z=-62x -2z=-6…

Q: (3) (a) Let T be a surjective linear transformation from P7 W. Show that dim(W) dim(W). Lastly, the…

A: Given below the detailed solution

Q: If A is a 4x6 matrix and T is a linear trạnsformation such that T: x Ax, then Select one: O a. The…

A: Topic- Linear Algebra

Q: List the eigenvalues of A. The transformation x→Ax is the composition of a rotation and a scaling.…

A: The given matrix, A=33-3333 We have to find the eigenvalues of A and rotation angle and the scale…

Q: Both B= {1, cost,cos t,cos' t} and C={1,cost,cos 2t, cos 3t} are bases of a vector space H = Span{l,…

A: Given bases are B={1, cost, cos2t, cos3t},C={1, cost, cos2t, cos3t} are bases of vector space…

Q: Let the matrix below act on C². Find the eigenvalues and a basis for each eigenspace in C². 45 -54…

A: Given, The matrix A = 45-54 We have to find the eigen values and eigenspaces.

Q: Let I be the line in R³ that consists of all scalar multiples of the vector w= -81- Find the…

Q: Describe what happens to xk as k→∞. Choose the correct answer below. O A. As k becomes sufficiently…

Q: [2 1 1. Find the inverse matrix using Gaussian Elimination for A=|1 1 1 li -1 2.

A: We know for a square invertible matrix [A:I]=[I:A-1] Now, [A:I]= 211 100111 0101-12 001 R1↔R2…

Q: Write the linear system in matrix form: dx/dt = -3x + 4y + e* sin(2t) dy/dt = 5x + 9z + 4et cos(2t)…

Concept explainers

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.

Question

Linear algebra

1. The given set with the given operations is not a vector space. Check all the properties and list which
properties fail to hold.
The set of all ordered pairs of real numbers with the operations (x,, y,) O (x,, y2) = (x, +x,,y, + y,) and
rO(x, y) = (0,0).
2. Prove that a vector u in a vector space has only one negative -u. [Hint: Assume that there are two
negatives and conclude that they must be equal.]
3. C(-00, 00) denotes the vector space of real-valued continuous functions.
Which of the following subsets are subspaces of the vector space C(-x,0) ? Explain.
a) All nonnegative functions.
b) All constant functions.
c) All functions f such that f(0) = 0.
d) All functions f such that f(0) = 5.
e) All differentiable functions.
4. a) Prove that proj (proj-v) = proj,v
b) Prove that proj (v – proj v) =0.
c) Explain parts a) and b) geometrically.
5. a) Prove that u-v2u + v for all vectors in R".
b) Explain parts a) geometrically.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step 5

Step by step

Solved in 5 steps with 5 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Linear Equations$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Let v1, v2, and v3 be three linearly independent vectors in a vector space V. Is the set {v12v2,2v23v3,3v3v1} linearly dependent or linearly independent? Explain.
Let u, v, and w be any three vectors from a vector space V. Determine whether the set of vectors {vu,wv,uw} is linearly independent or linearly dependent.
Let V be the set of all positive real numbers. Determine whether V is a vector space with the operations shown below. x+y=xyAddition cx=xcScalar multiplication If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
Find a basis for R2 that includes the vector (2,2).
Prove that in a given vector space V, the zero vector is unique.
Take this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. Prove that the set of all singular 33 matrices is not a vector space.
Prove that in a given vector space V, the additive inverse of a vector is unique.
Consider the vectors u=(6,2,4) and v=(1,2,0) from Example 10. Without using Theorem 5.9, show that among all the scalar multiples cv of the vector v, the projection of u onto v is the closest to u that is, show that d(u,projvu) is a minimum.
Let S={v1,v2,v3} be a set of linearly independent vectors in R3. Find a linear transformation T from R3 into R3 such that the set {T(v1),T(v2),T(v3)} is linearly dependent.
Use a software program or a graphing utility to write v as a linear combination of u1, u2, u3, u4, u5 and u6. Then verify your solution. v=(10,30,13,14,7,27) u1=(1,2,3,4,1,2) u2=(1,2,1,1,2,1) u3=(0,2,1,2,1,1) u4=(1,0,3,4,1,2) u5=(1,2,1,1,2,3) u6=(3,2,1,2,3,0)
Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector u=(1,1,1,1) in the form u=v+w, where v is in V and w is orthogonal to every vector in V.