Consider vector space V5 over field F with dimension 5. Consider linear mapping A E L(V5). Furthermore, assume that there are two mutually different non-zero vectors x, y E V5 satisfying Ax = Ay = 0. Then necessarily nul(A) = 2 rk(A) < 3 a · x + y E ker A, a E F

Consider vector space V5 over field F with dimension 5. Consider linear mapping A E L(V5). Furthermore, assume that there are two mutually different non-zero vectors x, y E V5 satisfying Ax = Ay = 0. Then necessarily nul(A) = 2 rk(A) < 3 a · x + y E ker A, a E F

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: 8.) What is the distance between an airplane flying at 120 feet above sea level and a turtle…

A: We find the distance between an airplane flying at 120 feet above sea level and a turtle swimming at…

Q: Is Mmxn(Q) is a vector space over R?

Q: Give an example of a nonzero vector field F such that curl(F) = 0 and div(F) = 0.

Q: Vz(x, y) V + y have the Show that all the vectors in vector field F same length, and state the…

A: The length or magnitude of a vector a, b is defined as follows: a, b=a2+b2 Note ca, b=ca, b, where c…

Q: Compute the vector assigned to the points P = (1, 1) and Q = (-3, -3) by the vector field F = (-y,…

Q: 2. For the vector field V = (7x³ + 2y2²)i + (4x²z - 7y²+3)j + (4y³ +2² -32²) k find the divergence…

Q: 1. Let f: R" → R be the function f(x) = cx+b for some fixed vector c ER" and scalar b. Show that f…

Q: Let r(t) =[x(t), y(t), z(t)]. Provide a definition for the continuityof the vector function r at a…

A: Define continuity of the vector function.

Q: ii) f x (g x h) is differentiable at t and d dt di {f x (g x h) } = (dg x h) +f x dt xh +fxg x dt dt

Q: vector field F = ⟨b cos(ab) + 2a + b, a cos(ab) + 2b + a⟩. State if it is conservative with reasons.

A: The given vector field If the vector field is conservative Then

Q: Determine the vector field of F. F(x, y) 3D yi — хј

A: Let us make a table to draw the vector field as shown in the image below.

Q: Let C₁ be the directed line segment that starts at (2,0) and ends at (-3,-5) and C₂ be the parabola…

Q: If A and B are vector fields, prove the following: V (A · B) = (B - V) A + (A · V) B+B× (V×A)+A ×…

Q: There are four questions about this problem in the test shown in random order. Out of 1000 customers…

Q: Determine whether each of the following vector fields F is path independent (conservative) or not.…

Q: Determine if the scalar vector field is Continuous on the indicated domain f(x) = (x²+1) S=(0,0) Х…

A: f(x)=x2+1/x is not Lipschitz continuous on the indicated domain.Explanation:Approach to solving the…

Q: Let F and G be vector fields with differentiable components. Express curl (F × G) in term of div and…

A: Let F and G be vector fields with differentiable components.Let We will compute .The cross product…

Q: For all differentiable vector fields v(r) and w(r), use index notation to show that - . - . v × (▼ ×…

A: We have to show .

Q: show that the given vector functions are linearly independent on (−∞,∞).

A: To show that the given vector functions are linearly independent x1t= et-et, x2t=etet

Q: 13. For the vector fields: A = x + 2y ŷ + 3z 2; B = 3y - 2x ŷ. Check both product rule given in…

Q: Define the terms sink and source in vector fields.

Q: magnitude 1.

A: To find : (x0,y0) Given: (x1,y1) = (2,1) Magnitude = 1

Q: Find the domain of the vector functions, r(t), listed below. You may use "-INF" for -∞o and use…

A: According to bartleby guidelines we supposed to do the first one of multiple questions kindly repost…

Q: 1. For each of the following vector fields F(1, y), decide if there is function f(z, y) such that F…

Q: 1 ²+1 (a) Find the inner product (f. 9) = (b) Verify the Cauchy-Schwarz Inequality. 2. Let f(x)= x…

Q: x-z+4 8x-y+3z² 3. Given vectors field G = 3* 4+z2 :ax + y3+z - ay and a position vector, r, = 2ax –…

Q: iii) (f x g) xh is differentiable at t and d dt 7 {(f x g) x h} xgx h + x dt dg xh +(fx g) x dt dt

Q: Determine whether or not F(x,y) = (xy + y²,x² + 2xy) is a conservative vector field.

A: Given vector field is Fx,y=xy+y2,x2+2xy Fx,y=F1,F2 A vector field is said to be conservative if…

Q: Let x1(t), x2(t), . . . , xn (t) be vector functions whose ith components (for some fixed i )…

A: Given that x1(t), x2(t), . . ., xn(t) be vector functions whose ith components ( for some fixed i)…

Q: In a rectangle, the width is one-third the length. The total perimeter is 40 inches. Find the length…

A: Solution: Given: Width is one third of the length of rectangle and the perimeter of rectangle…

Q: 1) Given the vector fields F1 = x²2 and F2 = xâ+ yŷ + z2, calculate the divergence and curl of F1…

Q: Sketch the given vector fields by drawing some representative vectors. (a) F(x, y) = (x − y) î+xî…

A: Given: a F→x, y=x-yi^ +xj^b F→x, y=y, -xx2+y2c F→x, y=2yj To sketch the given vector field.

Q: If φ and ψ are smooth scalar fields, show that ∇×(φ∇ψ)=∇φ×∇ψ

Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

Consider vector space V5 over field F with dimension 5. Consider linear mapping A E L(V5).
Furthermore, assume that there are two mutually different non-zero vectors x, y E V5 satisfying
Ax = Ay = 0. Then necessarily
nul(A) = 2
rk(A) < 3
a · x + y E ker A, a E F

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 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Vector Space$

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

(b) Show that the vector field, F3 yz î + zx ŷ + xy 2 can be written both as the gradient of a scalar and curl of a vector. Find the scalar and vector potential for this function.
Let A(t) = f(t)i + f₂(1) and B(t) = g₁(t)i + g₂(t)},t = [0, 1], where f₁f2, 81, 82 are continuous functions. If A(t) and B(t) are non-zero vectors for all and 7(0) = 2î +3ĵ, A(1) = 6î +2), B(0)=31 +23 and 6i- A B(1) = 27 +6, then show that A(1) and B(t) are parallel for some t.
4. Assume A is a constant vector field and R = xi + yi + zk. (a) Simplify the expression (R. VR) · Ã+ V ·V × V × ((Ã × R) × (R × Ã)). (b) Assuming A the position vector of a point on the unit sphere centered at the origin, evaluate the above expression at R = A.
Verify Green's theorem for a vector field F(x, y) = y²³i+z²j and a triangle bounded by the lines z + y = 1 and -z+y = 1 and y= = 0. (Hint: all the conditions in the Green theorem must be verified.)
Show that the vector field A (3x²z + y?)ax + 2xya, + x³a, is conservative.
Let the vector C represents the coefficients of a polynomial pt=c1+c2t+c3t2+c4t3. Express the conditions p0=1, p'0=2, p1=1, p'1=0 in the form AC=b.
The vector field F(x,y)= can be written as: O1. F(x,y)=(xy+x2 )i - (8)j O II. F(x,y)=(xy+x² )i + (2-10)j O II. (x,y)=(xy-2)i + (x² -10)j O IV. F(x,y)=x(y+x)i + (2 -10)j
Q5: a) Show that the vector field F = (x² - y² + x)i- (2xy + y)j, is irrational.
(a) For all vectors ù, v, w € R³, ủ x ( xử) = ( xử) xử. True False (b) Let F: R³ True False R³ be a vector field of class C². Then div(curl F) = V. (V x F) = 0.
G is a vector field defined as G = Bxlnzi + Dy zj + (+ y³) k. b) If C is the straight line from (1,1,1) to (2,1,2) find S, 2xlnzdx + 2y²zdy +y dz.
Let A(t) = f(t)i + f₂(1) and B(t) = g₁(t)i + g₂(t)},t = [0, 1], where f₁f2, 81, 82 are continuous functions. If A(t) and B(t) are non-zero vectors for all and 7(0) = 2î +3ĵ, A(1) = 6î +2), B(0)=31 +23 and 6i- A B(1) = 27 +6, then show that A(1) and B(t) are parallel for some t.