Let V, W be finite-dimensional vector spaes over C.(a) If T ≤ L(V, W), prove the Rank-Nullity Theorem with T.(b) If V = Mnxn is the space of n - n matrices over C and A, B E Mnxn, proverank(AB) ≤ min{rank(A), rank(B)},rank(A + B) ≤ rank(A) + rank(B).

Answered: Image /qna-images/answer/01802a5a-b867-4341-a5f2-f9f421f98d53.jpg

Let V, W be finite-dimensional vector spaes over C. (a) If T ≤ L(V, W), prove the Rank-Nullity Theorem with T. (b) If V = Mnxn is the space of n x n matrices over C and A, B E Mnxn, prove rank(AB) ≤ min{rank(A), rank(B)}, rank(A + B) ≤ rank(A) + rank(B).

Let V, W be finite-dimensional vector spaes over C. (a) If T ≤ L(V, W), prove the Rank-Nullity Theorem with T. (b) If V = Mnxn is the space of n x n matrices over C and A, B E Mnxn, prove rank(AB) ≤ min{rank(A), rank(B)}, rank(A + B) ≤ rank(A) + rank(B).

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 P2(R) be the vector space of polynomials over R up to degree 2. Consider B = {1+x= 2x°, -1+a…

A: given B=1+x-2x2 , -1+x+x2 , 1-x+x2B'=1-3x+x2, 1-3x-2x2 , 1-2x+3x2 b) find the coordination matrices…

Q: An inner product and an associated norm are defined on R² by the weighted scalar product where u =…

Q: (a) {(a, b, c) ER³ | 2a = 3c}. (b) {(a, b, c) ER3 | a = 0, b = 3c}. (c) {(a+b, ab, 2a + b) | a, b R}…

A: (a), (b) , (c), (d) are subspaces .

Q: Which one of the following is a subspace of the vector space M2 (R) of 2 x 2 matrices? ss (X) the…

A: .

Q: If the matrix of change of basis form the basis B to the basis B' is A= change of basis from B to B…

Q: (i) Consider the vector space M2×2 with the trace inner product ( A,B) = trace(A'B) and let A = .…

A: We are entitled to solve only one question at a time. Given - To find -

Q: 2. Let A be an m by n matrix and B be an n by n matrix. (a) If rank(A) = nullity(A), prove that n is…

A: Since you have posted multiple questions with multiple sub parts, we will provide the solution only…

Q: V is a vector space with bases B = (v1, V2, v3) and B' = (w1, w2, w3). The change of basis matrix 1…

Q: Which of the following sets is a basis for the subspace U={(a,b,c,d)∈R4 ∣ a−b=0,a+d=0,b+d=0} in the…

Q: What is the relationship between the kernel of a linear transformation from a finite dimensional…

Q: Show that any isometry of R^n which fixes the origin is linear, and hence must be of the form f(x) =…

A: Introduction A square matrix is said to be an orthogonal matrix if…

Q: Let the basis {(1,1,0), (0,1,1), (1,0,1)} of the space R^3 be orthonormal in the scalar product…

A: {Here we use a•(b+c) = a•b +a•c}

Q: Let X,Y be a finite dimension vector space then O dim(x)=dim(Y) O dim(X)<dim(Y) O dim(X)zdim(Y) O…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Which of the following is a basis for the subspace {:) X y z x = y+z and x, y,z €RC M2x3(R) ? 0 x 0…

Q: let W be an inner product space and let w1 and w2 be vectors in W. suppose ll w1 ll =5^1/2 and ll w2…

Q: a) Give an example of subspaces U = span(f1, f2, f3) and V = span(g1,92; g3) of RaJa] such that…

Q: Let I2 ∈ M2(K) be the 2 × 2 identity matrix, where K is a field. For any A ∈ M2(K), let v1 =I2, v2…

A: The objective of the question is to determine the linear dependence or independence of the given…

Q: Find the dimension of the following vector spaces. (a) V = {0}: dim V = (b) V = R¹: dim V- (c) V is…

A: (.) Dimension: The number of vectors in a basis of a vector space is called dimension of..(.)…

Q: 8. = - Prove that the set is a basis for the vector space of 2x2 matrices

Q: d. Find the transition matrix from C to B. e. Find the coordinates of u = (2, –2) in the ordered…

Q: 20) Let W be the vector space of all symmetric 2 x 2 matrices. Show that W sisomorphic to R³

A: Introduction: The set of vectors v1, v2, ..., vn in V is said to be linearly independent if the only…

Q: 2. A 2D-subspace (manifold) in 3D- space is defined by the two vectors d;=é,+ê,-2êzand ā,=ê,+2êz-3ê,…

A: A 2D-subspace (manifold) in 3D - space is defined by the two vectors a→1=e^1+e^2-2e^3 and…

Q: Let V = R' and define vector addition as (x, y, z)+(x', y', z')= (x+x',y+y',z+z') (standard…

Q: Verify all vector space axioms under the given operations (x,y,z)+(x',y'z') = (x+x'+1,y+y'+2,z+z'+3)…

A: The objective of the question is to verify if the given operations satisfy all the axioms of a…

Q: 2 COLAS -1 Consider the two bases B= for R² and C= P √2 a.) Compute the change-of-coordinates matrix…

Q: Let A be a 9x6 matrix with nullify equal to 1 a) find the rank and nullify of A^T b) suppose that…

Q: Consider the three vectors 3 --)--(C)--0) 3 W = 9 -3 u= -2 3 Let S denote the one dimensional…

A: As per the question we are given three vectors u, v, w and S is the one dimensional subspace spanned…

Q: Let W be the subspace of IR' spanned by the vectors 1 4 -2 and -1 -7 Find the matrix A of the…

Q: 3. Let P, be the vector space of polynomials of degree at most 2. Define a map D : P2 → P, it sends…

Q: Let V be an F-vector space and A, B, C are subspaces of V. (a) Show that the equation An (B+C) =…

Question

Let V, W be finite-dimensional vector spaes over C.
(a) If T ≤ L(V, W), prove the Rank-Nullity Theorem with T.
(b) If V = Mnxn is the space of n - n matrices over C and A, B E Mnxn, prove
rank(AB) ≤ min{rank(A), rank(B)},
rank(A + B) ≤ rank(A) + rank(B).

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: Divide the proof into three cases.

VIEW

Step 2: Simple explanation

VIEW

Step 3: Required result

VIEW

Solution

VIEW

Step by step

Solved in 4 steps with 4 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

Could you explain (b) as well?

Solution

by Bartleby Expert

SEE SOLUTION

Similar questions

plz provide answer for q12 asap
V is a 3-dimensional vector space with basis B = (vı, V2, v3). Let w = vi + v2 + v3, w2 = v2+V3, W3 = v1 +v2+2v3 and set B' = (w1, w2, W3), which is also a basis of V. a) Determine the change of basis matrix from B' to B, Pg-→B. b) Determine the change of basis matrix from B to B', P3¬B'.
4) Let V be the vector space of polynomials of degree <2 and let p(x) = ax² + bx + c. Find the adjoint operator (T*@)(p) for T:V → V and o:V → R given by d) Tp(x) = x²p(x) + x³p'(x), @(p) = p"(1).
Prove that the set of all matrices, under the usual operations is NOT ALWAYS a vector space. Provide a case/example.
Let A be any 5x7 matrix for which the col(A) has dimension 3, calculate: the nullity(A), and, state which vector space R^k that null(A) is a subspace of (give k). O A. nullity(A)=2, k=7 B. nullity(A)=4, k=7 OC. nullity(A)=2, k=5 O D. nullity(A)=4, k=5
Help me please
If W\index{1} and W\index{2} are subspace of a vector space V(F), then show that W\index{1}+W\index{2} is also a subspace of V(F).
(3) Let W be the subspace of R³ with orthogonal basis {u₁, u₂}, where 1 --0---0 = 1 -2 Let = -6 y = -3 1 2 Write y as the sum of a vector in W and a vector in W¹.
Apply Gram-Schmidt to (1, 0, −1, 1), (2, 1, −1, 0) , (2, −1, −1, 3) to obtain an orthonormal set with the same span as these vectors.
Solve all quation
C: Let T be the linear operator associated in the standard basis B of F2 with the matrix 2 - [44] - 1 Let B' be the basis ((2, -1), (2, 1)) M = b) Using eachof these three matrices find the dimension of the range of T (you must check that the dimension is the same with all the matrices)
Let A be any 5x7 matrix for which the col(A) has dimension 3, calculate: the nullity(A), and, state which vector space R^k that null(A) is a subspace of (give k). A. nullity(A)=4, k=7 B. nullity(A)=2, k=5 C. nullity(A)=4, k=5 D. nullity(A)=2, k=7