Please just answer (a,b &c) Question This question here is to show that linear maps out of/into direct sums "behave" like matrices. Let V1,...,Vn, W1,..,Wm be vector spaces over F. (a)(b)Suppose we are given linear maps Tij: VjT: V₁0V₂ → W₁ ...Vnthat "behaves" like the mx n matrix of linear mapslinear mapT₁1T12T21 T22Tm1 Tm2Show that every linear mapT: V₁0 Vnwith coefficients Tij € L(V₁, W₁)→Conclude that the set of linear mapsT: V₁0 V₂can be identified with the set of matricesT11T12T21 T22Tm1 Tm2is of this form for linear maps Tij: V; → W; (hint: inclusion into direct sum/projectiononto subspace)(c)TinT2nW₁ ... WmTmn......W; for each i,j. Construct aWmW₁ ... WmTinT2nTmn(d)Does anything familiar happen when V₁Vn = W₁==(Hint: when Vj = W₁ = F what is L(V₁, W;) isomorphic to? Think dimensions....)=...Wm= F?

We shall solve first question only as you have asked more them one different question. For others…

(a) linear map Suppose we are given linear maps Tij: Vj W; for each i,j. Construct a T: V₁0 V₂ W₁ ... Wm that "behaves" like the mx n matrix of linear maps T11 T12 … T21 T22 Tm1 Tm2 Tin T2n Tmn

(a) linear map Suppose we are given linear maps Tij: Vj W; for each i,j. Construct a T: V₁0 V₂ W₁ ... Wm that "behaves" like the mx n matrix of linear maps T11 T12 … T21 T22 Tm1 Tm2 Tin T2n Tmn

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.6: The Matrix Of A Linear Transformation

Problem 22EQ

See similar textbooks

Related questions

Q: Prove that for all natural number n, 52n-1 is divisible by 8.

A: Will prove the result by mathematical induction

Q: Check if the following statement is TRUE or FALSE. Let p(x) and g(x) be two open predicates in a…

A: We will check whether the given expressions are equivalent or not

Q: 7. The solution z = (x, y) to the equation in Problem 6 is A. /1 (12.+ 1/3) 2 √3 B. (-12. ± 12³) +…

A: 7. Given equation is z2 + z + 1 = 0

Q: Construct the LP model then solve. Show complete, neat, and orderly solutions. This time, our…

A: Given- This time, our immune system is the best defense. With this, a Melagail wishes to mix two…

Q: Solve the differential equation x²(x)² + 2xy-15y² = 0. Ein th diffa atial equation

Q: The average of an electrician's hourly wage and a plumber's hourly wage is $32. One day a contractor…

A: Given: Average of hourly wages of both electrician and plumber is $32. The total wage on hiring…

Q: Find the general solution of the differential equation y" + 2y + 5y = 2 sin(2t). NOTE: Use c₁ and ce…

Q: 3. In evaluating a double integral over a region D, a sum of iterated integrals was obtained as…

A: The given integral is ∬Df(x,y)dA=∫01∫02yf(x,y)dxdy+∫13∫03-yf(x,y)dxdy. (i) We need to sketch the…

Q: Problem 1: Solve the following assignment problem shown in Table using Hungarian method. The matrix…

A: As per the question we are given a 5×5 matrix which we have to solve (find the optimal solution)…

Q: (b) Show that U (x, y) = eu cos v, V(x, y) = eusin v are conjugate harmonic of each other if f = u +…

A: It is given that, f = u+iv is analytic. We have to show that, U(x, y) = eu cos v and V(x, y) =…

Q: 2. Let (2n) and (yn) be Cauchy sequences of real numbers. Define (Tn) to be equivalent to (yn),…

A: Given : xn and yn be the Cauchy sequences of real numbers. And (xn) ~ (yn) if limn→∞xn-yn = 0 To…

Q: Set up an iterated double integral equal to the volume of the solid in the first octant bounded only…

A: The given problem is to set up an double integral that gives the volume of solid bounded by given…

Q: Use Bairstow's method to determine the roots of f(x) = 9.34 - 21.97x + 16.3x² - 3.704x³ r = -1 and…

A: We need to determine the roots of the equation: fx=9.34-21.97x+16.3x2-3.704x3 using Bairstow's…

Q: Draw the Taxi-metric hyperboloid |x| + |y| + |z| = 1

A: As per the question we are given the equation of Taxi-metric hyperboloid as : |x| + |y| + |z| = 1…

Q: Perform error analysis of approximation derived (a) 1 x² 1 - at xo = 0 X (b) sin (₁) ₁ X- at x = = 0…

A: Given that: a 1x2-1 at x0=0b sin xx-1 at x=0c exp x2-1 at x=1 and x=-1 The objective is to analyse…

Q: Set up an iterated double integral equal to the mass of the lamina in the shape of the region inside…

A: Mass of a lamina is given by M=∫∫A δx,y dA

Q: Let A be a 3 x 2 matrix with lineary Independent columns. Suppose we know that Al-5. Find a solution…

Q: Y= x^5-2x^4+3x^2-5x+1 find the c and y intercept

A: X-INTERCEPTS Plug y=0 into the equation and solve the resulting equation x5−2x4+3x2−5x+1=0 for x.…

Q: 0.5 a= 5 -0.5 2 For the above plot of the ellipsoid ()² + (†)² + (-)². a, b and c are positive…

A: Answer from the figure.

Q: Exercise: 6 1. (dy/dx)-2y tan x=y² tan²x. Ans. (-1/y) sec² x=c+ (1/2)x tan³x

Q: a)!^ 1. Evaluate the integral 4x - 1 (x + 2)(2x − 1) dx. - la

A: As these questions are not interrelated, these are considered as different questions. As per our…

Q: 1. (A population model) Consider the differential equation dP dt -P(2P-1)(P-3) where P(t) is the…

A: Given differential equation is dPdt=19P(2P-1)(P-3), where P(t) is the population (in thousands) at…

Q: Solve the following system using GS iteration method use x0 = (0000) 10x₁ - x₂ + 5x3 = 6 -x₁ + 11x₂…

A: The Gauss Seidel method is an iterative method to find the approximate solution of the system of…

Q: A scientist is doing an experiment on the growth of the corona virus. He started the experiment with…

A: The model is provided as: dPdt=aP1-PK. We need to use this model and the conditions provided to…

Q: 3- Show that the local truncation error at the pint (ih.jh) of the Crank-Nikolson approximation to…

Q: 20 23 12 25 14 27

A: Given as,

Q: A motion has R++) = (3.8m (347) 2² + (3 cos (3+));²³² + 3 +K A at time + -4 is. A vector dan gend to…

Q: Consider the function f(x) = and the data (2,0.25), (3,0.167), (4,0.125) 2x 1) Find Lagrange…

A: 1] Lagrange Interpolating polynomial formula for the data x 2 3 4 y 0.25 0.167 0.125…

Q: In each of Problems (2) through (9), find the general solution of the given system. Calculations are…

Q: The determinant of a 2 by 2 matrix A is equal to the area of the parallelogram with two adjacent…

A: Given the 2 by 2 matrix A. We have to find the determinant of a matrix A is equal to the area of…

Q: Let p denote the proportion of students at a large university who plan to use the fitness center on…

A: Solution: Given that, Ho : P = 0.5 Ha : P > 0.5 Now, we have to find the values of P (a) ZSTAT =…

Q: Q7 Define f(x)=³-3x+1, and o(a)=x²-2. The following information will be useful: f is irreducible…

A: According to the guidelines, we are required to solve the first two sub-questions. Kindly, re-post…

Q: Draw the planar representation of the graph below

Q: An initial value problem for a second order linear differential equat is given as follows: 3t…

A: We have to determine the longest interval of t in which the initial value problem has an unique…

Q: Solve the following system using GS iteration method use x0 = (0000) 10x₁x2 + 5x3 = 6 -x₁ + 11x₂ -…

A: To find- Solve the following system using GS iteration method use x0 = 0, 0, 0, 0 10x1 - x2 + 5x3 =…

Q: Consider the two differential equations: I: (4x + 3y²)dx + 2xydy = 0 2y II: (x²+²x) dx + (In x² −…

A: Exact Differential equation: "The given differential equation of the form…

Q: (a) (3) Let V be a vector space over some field F Let W be a subspace of a vector space V. Suppose…

A: It is given that V is a vector space over the field F. a) Suppose W is a subspace of a vector space…

Q: (77) (04-70° + 180² - 200+ 200+8) V = 0

Q: Find the general solution of the differential equation y" - 4y - 12y -20t+24t². =

Q: اعه W = {4; [0₁+0) IR is absolutely continuous what is absolutely continuous ?

Q: In a dance performance, four dancers form a diamond with vertices A(2, 0), B(0, 2), C(-2, 0), and…

A: The vertices are provided as 2, 0, 0, 2, -2, 0, 0,-2. The translation vector is 0, 4. We need to…

Q: Theorem Proof Let f: G→→ H be a group homomorphism. Then, Im fs H. The kernel of f, we write Ker f,…

Q: Let & = (A xEA) be a family of sets, and let B be a set, Then B- (UA) = U (B-A). «ΕΔ a EA True False

Q: 1. Determine the Inverse Matrix of the following using the 5 steps and Gauss - Jordan Method. i. A =…

A: INTRODUCTION: The row reduction approach for solving systems of linear equations is the name given…

Q: Consider a function f(x)and its second Taylor polynomial P2 (x) centered at a .if p2x is not…

A: As per the question we are given a function f(x) and its second order Taylor polynomial P2(x)…

Q: To get a B in math, Alexandria Pappas must average 80 on five tests. Scores on the first four tests…

A: Given Alexandria Pappas score in first four test are 84, 76, 84 and 78.

Q: A) Solve the IVP: (2x - 4)- - 2 ln 5x + 2y = 0, y(3) = 3. dy dx B) Give the largest interval over…

Q: Find the direction of maximum change of f(x, y, z) = xey + z² at (1, In 2,2).

A: Introduction: A function's maximum directional derivatives at a particular point P are produced in…

Q: If W is a subspace of R and v is any vector in R", then ||projwv|+|v-projwv||2= |v||2.

A: Introduction: Every vector that doesn't lie in a subspace can be uniquely expressed as a sum of two…

Q: fermat's theorm states, if p is prime & a is a positive integer not divisible by p then-------- and…

A: Fermat's states that if p is a prime and a is a positive integer not divisible by p, than ap-1≡1 mod…

Question

100%

Please just answer (a,b &c)

Question

This question here is to show that linear maps out of/into direct sums "behave" like matrices.

Let V₁,...,V_n, W₁,..,W_m be vector spaces over F.

(a)
(b)
Suppose we are given linear maps Tij: Vj
T: V₁0
V₂ → W₁ ...
Vn
that "behaves" like the mx n matrix of linear maps
linear map
T₁1
T12
T21 T22
Tm1 Tm2
Show that every linear map
T: V₁0 Vn
with coefficients Tij € L(V₁, W₁)
→
Conclude that the set of linear maps
T: V₁0 V₂
can be identified with the set of matrices
T11
T12
T21 T22
Tm1 Tm2
is of this form for linear maps Tij: V; → W; (hint: inclusion into direct sum/projection
onto subspace)
(c)
Tin
T2n
W₁ ... Wm
Tmn
...
...
W; for each i,j. Construct a
Wm
W₁ ... Wm
Tin
T2n
Tmn
(d)
Does anything familiar happen when V₁
Vn = W₁
==
(Hint: when Vj = W₁ = F what is L(V₁, W;) isomorphic to? Think dimensions....)
=...
Wm= F?

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps with 1 images

SEE SOLUTION Check out a sample Q&A here

Follow-up Questions

Read through expert solutions to related follow-up questions below.

Follow-up Question

Ok can you please work on parts (b &c)

Question

This question here is to show that linear maps out of/into direct sums "behave" like matrices.

Let V₁,...,V_n, W₁,..,W_m be vector spaces over F.

Solution

by Bartleby Expert

SEE SOLUTION

Similar questions

Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).
In Exercises 1-12, determine whether T is a linear transformation. T:MnnMnn defines by T(A)=AB, where B is a fixed nn matrix
Find a basis B for R3 such that the matrix for the linear transformation T:R3R3, T(x,y,z)=(2x2z,2y2z,3x3z), relative to B is diagonal.
In Exercises 1-12, determine whether T is a linear transformation. 5. T:Mnn→ ℝ defined by T(A)=trt(A)
In Exercises 1-12, determine whether T is a linear transformation. 4. defined by , where B is a fixed matrix
Here is a data matrix for a line drawing: D=[012100002440] aDraw the image represented by D. bLet T=[1101]. Calculate the matrix product TD, and draw the image represented by this product. What is the effect of the transformation T? cExpress T as a product of a shear matrix and a reflection matrix. See Problem 2. 2. Verify that multiplication by the given matrix has the indicated effect when applied to the gray square in the table. Use c=3 in the expansion matrix and c=1 in the shear matrix. T1=[1001] Reflection in yaxis T2=[100c] Expansion or contraction in ydirection T3=[10c1] Shear in ydirection
Let T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.
The matrix A = " 1 0 0 2 # is a linear map from R 2 to R 2 . Draw the modified shape of the circle x 2 + y 2 = 1 after applying A on R 2
Give the standard matrix of the linear transformation T:R2→R2 that firstreflects points through the line x2=x1 and then expands R2 in the horizontal direction by afactor of 5.
Hey , how do i show, for an n × n matrix A (or the linear map generated by it), the following properties are equivalent. a)A is inject b) ) det(A) ̸= 0 Thank you in advance
Let the matrix reprentation of a certain linear mapping T be