5 (a)indicateu vectors:Let L: R2 → R³ be a lincar map, having the following effect on theL(1, –4) = (2,0, -1) and L(0,3) = (0, 1, 1).Compute L(1,0).(b)Find the image and kernel of this map L, and verify the equalitydim R?= dim Ker L+ dim Im L.

Answered: Image /qna-images/answer/c14a34f8-2a51-4ea8-9506-8f81457aa403.jpg

Answered: 5 (a) Let L: R2 → R be a lincar map,…

5 (a) Let L: R2 → R be a lincar map, having the following effect on the indicateu vectors: L(1, -4) = (2,0, –1) and L(0,3) = (0, 1, 1). Compute L(1,0).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: 2. Let v, = (1,2,1), v2 = (2,9,0), v3 = (3,3,4). Show first that S = {v1, V2, V3} %3D is a basis for…

A: 1. A finite set of vectors v1,v2,...,vn of a vector space V over a field F is said to be linearly…

Q: Find the dimension of the vector space U of all linear transformations of V into W for each of the…

A: a)- Given: V=R2 , W=R3 Since R2 is 2-dimensional vector space and R3 is 3-dimensional vector space,…

Q: 2. Let V be an n-dimensional vector space and T: V→V be defined by T(v) = 2v. Find the kernel,…

A: Since you have posted multiple question we will solve the first question for you. If you want any…

Q: (a) If g0 = 4, is the solution function increasing or decreasing? Illustrate your reasoning on the…

A: Define increasing and decreasing:

Q: (1' '3' Consider vị = | 2 V2 = 9 and v3 = 3 ) as vectors in R3. 4 Let T: R3→R2 be a linear…

Q: Consider the data: (zo, f(xo)), (₁, f(x1)), (2, f(x₂)), (#3, f(#3))).\\ The Lagrange basis…

A: Lix=x-x0x-x1·..x-xi-1x-xi+1·...x-xnxi-x0xi-x1·..xi-xi-1xi-xi+1·...xi-xn∴Lixj=0 for i≠j1 for…

Q: 6. Let p(t) = ² + 1 and g(t) = t - 4 be vectors in P2 with inner product (f(t), g(t)) = f(t)g(t) dt.…

Q: Consider the four maps on R² given by Tix = Ax+b; for i = 1,2,3,4, where A = [ and the vectors b,…

A: The main objective is to show that fixed points of T’s form the vertices of a square S. Also draw…

Q: Let 1 A linear transformation T; from R to R° maps vectors v = (V1, V2) to v' = (v, , v) according…

A: Given

Q: 3. Consider the linear transformation 7" : R²R where for a vector z = Hille T** (x)= 22 4 Find a…

Q: 8. Let V={(x,y)x,y € R}. Define addition on V as follows: (x»x)+(x,y)=(x +2x, 1,y+Y,+2) Find the…

A: To find zero vector of V

Q: Let x = (1, 1, 1, 1)T and y = (8, 2, 2, 0)T . Verify that x − p is orthogonal to p.

Q: 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…

A: The matrix A=1002 is a linear map from ℝ2 to ℝ2 . To Draw: The modified shape of the circle x2+y2=1…

Q: Let L : R → R² be defined by L(æ) = L T2 (:) - () Show that L is not a linear transformation by…

A: We have to prove that L is not a linear transformation.

Q: 2: Show that a map T is linear if and only if T(Au + v) = XT(u)+ T(v) for all A ER and vectors u, v.

A: A linear transformation satisfies T:U-V Then T(u+v) = T(u) +T(v) T(au) = aT(u) For all a…

Q: Determine the values of y such that D(9,3,-2), E(3,4,7) and F(21,y,-20) are collinear

Q: ent, find a linear relation. a) vi = (2, 1, 0), V2 = (0, 1,0), V3 = (-1,2,0) %3D b) wi = (1,2, –2),…

Q: Let V be a vector space and let v, v2, vz e V. Suppose f : V → R³ is a linear transformation and fG)…

A: Given: f:V→ℝ3 is a linear transformation fv1=-5-45, fv2=-4-4-2, fv3=-12-4 a. As f is a linear…

Q: #2. Let P be a parallelogram, having area 6 square units, spanned by two linearly independent…

Q: Show that r|(1) and r2(1) define the same line, where rị(t) = (3, –1, 4) + t (8, 12, –6) r2(1) =…

A: Two lines have the same direction if their direction vectors v1→ and v2→ are directly proportional.…

Q: Let a, b and c be the first three non-zero digits of your Banner ID number and consider the points P…

A: Due to Bartleby answering guidelines I have solved only first three Subparts of the given question…

Q: 1 (b) Find the span of vectors i = |0 and j = |1 in R3

Q: Parts of this question are independent (a) Suppose īj, ūz ā, b,č 4, det[ū, ī2 č =-1. Find det [v, ūz…

Q: Let V be a vector space and suppose there exist maps S, T = L(V, V) such that range(S) ℃ null(T).…

A: Let V be a vector space and suppose there exist maps S,T:V→V such that .We need to prove that the…

Q: 33. Let V - P3 be the vector space of polynomials of degree <3, and let p₁(x) = (2-x)² and p₂(x) =…

Q: Let V₁, V₂ be vectors in R³ given by ---- V1 a) Find a vector w R³ with the following properties: •…

Q: 1. Given vectors B = (1,–2,3,0),C =(-2,1,0,4) in R*. (a) Examine if vector X = (-4,-1,6,12) is in…

A: According to our guidelines we can answer only one question and rest can be reposted.

Q: Let V₁, V2, and v3 be vectors in a vector space V, and let T:V→ R³ be a linear transformation for…

A: Given :T(v1)=(1,-1,2)T(v2)=(0,3,2)T(v3)=(-3,1,2)

Q: Let v1 = (1,2,1), v2 = (2,9,0), v3 = (3,3,4) . Show first that S= {v1, v2, v3} is a basis for ℝ^3.…

A: A linear map is a mapping A to B between two vector spaces that preserves the operations of vector…

Question

5 (a)
indicateu vectors:
Let L: R2 → R³ be a lincar map, having the following effect on the
L(1, –4) = (2,0, -1) and L(0,3) = (0, 1, 1).
Compute L(1,0).
(b)
Find the image and kernel of this map L, and verify the equality
dim R?
= dim Ker L+ dim Im L.

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 3 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Linear Transformation$

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

Lett: V→ V' be a linear transformation. Then (1) Ker (t) is a vector sub space of V, and (ii) im (1) is a vector sub space of V
Let: R² R³ be a linear map such that /(1,2)=(1,1,2), and /(2,1)=(2,2,1). Then which equalitie are correct? f(0,0) = (0,0,0). ƒ(1, 1) = (1, 1, 1). Of(-1,1)=(-1,-1,1). Of(-1,-1)=(-1,-1,1).
(ii) Let -2 v = 3 W = span{ |} and The orthogonal projection of v onto W is projw(v) = and the component of v orthogonal to W is perpw(v) =
(1) Prove that any linear transformation T : R" → R" maps a line passing through the origin to either the zero vector or a line passing through the origin. Generalize this for planes and hyperplanes. What are the images of these under linear transformations?
4. Let B = {v1, v2, V3} and C = {w1, w2, w3} be bases for R3, with vectors defined below. -(:) -() --() v1 = V2 = V3 = and --(:) --E) --() 1 wi = , w2 = 1 , W3 = 1 Let L: R³ → R³ be the linear transformation defined by L: Find [L]B and [L]c, the matrices associated to L with respect to B and with respect to C.
7. Consider the set of vectors S = {u = (-1, 2, – 1), v = (2, – 5, 0), w = (1, – 3, – 1)}, then find the vector orthonormal projection of "u" on "v" and the vector orthonormal projection of "u" on "w" respectively.
8. Let A = | 1 -3 -5 -7 7 5 (a) Find the image of u = and define T: R³ R2 by T(x) = Ax. 2 1 under T. (b) Find a vector x whose image under T is b = -12 12 Explain why your work makes sense.
4. Let T: U→V be a linear transformation from a vector space U (F) into a vector space V(F) and U(F) be finite dimensional. Show that U and R(T) have the same dimension iff T is nonsingular.
6. Determine the matrix of the linear mapping defined by first mirroring the space vectors in the line with equation (x, y, z) = (2-2+, -2+2+, 1-t) and then rotating 풍 them by the angle II around the x-axis in the positive direction.