Consider the vector space P2 := P2(F) and its standard basis α = {1,x,x^2}. 1Prove that...

Consider the vector space P₂ := P₂(F) and its standard basis α = {1,x,x^2}.

1Prove that β = {x−1,x^2 −x,x^2 + x} is also a basis of P₂

2Given the map T : P2 → P2 deﬁned by T(a + bx + cx2) = (a + b + c) + (a + 2b + c)x + (b + c)x²
compute [T]^β_α.

3 Is T invertible? Why

4 Suppose the linear map U : P2 → P2 has the matrix representation

(1 0 0

0 2 0

0 0 4)

Compute [UT]^α_α and complete the following formula
(UT)(a+bx+cx²) =

Solutions

Expert Solution

Here we use the theory of verifying a basis or not. Matrix representation of linear transformation and coordinate basis representation etc. The last part provides some incomplete data.

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Consider the vector space P2 of all polynomials of degree less than or equal to 2...

Consider the vector space P2 of all polynomials of degree less than or equal to 2 i.e. P = p(x) = ax + bx + c | a,b,c €.R Determine whether each of the parts a) and b) defines a subspace in P2 ? Explain your answer. a) ( 10 pts. ) p(0) + p(1) = 1 b) ( 10 pts.) p(1) = − p(−1)

Let V be the vector space of 2 × 2 real matrices and let P2 be...

Let V be the vector space of 2 × 2 real matrices and let P2 be the vector space of polynomials of degree less than or equal to 2. Write down a linear transformation T : V ? P2 with rank 2. You do not need to prove that the function you write down is a linear transformation, but you may want to check this yourself. You do, however, need to prove that your transformation has rank 2.

5. With each stated property, give an example of the 2-space vector field F(x,y). (a) F...

5. With each stated property, give an example of the 2-space vector field F(x,y). (a) F has a constant direction, but ||F|| increases when moving away from the origin. (b) F is perpendicular to the vector field and ||F||=5 everywhere.

'(16) Consider the map T : P2(F) →P2(F) deﬁned by T(p(x)) = xp'(x). (a) Prove that...

'(16) Consider the map T : P2(F) →P2(F) deﬁned by T(p(x)) = xp'(x). (a) Prove that T is linear. (b) What is null(T)? (c) Compute the matrix M(T) with respect to the bases B and B' of P2(F), where B = {1,1 + x,1 + x + x2}, B' = {2−x,x + x2,x2}. (d) Use the matrix M(T) to write T(7x2 + 12x−1) as a linear combination of the polynomials in the basis B'.

1. Let ? be a finite dimensional vector space with basis {?1,...,??} and ? ∈ L(?)....

1. Let ? be a finite dimensional vector space with basis {?1,...,??} and ? ∈ L(?). Show the following are equivalent: (a) The matrix for ? is upper triangular. (b) ?(??) ∈ Span(?1,...,??) for all ?. (c) Span(?1,...,??) is ?-invariant for all ?. please also explain for (a)->(b) why are all the c coefficients 0 for all i>k? and why T(vk) in the span of (v1,.....,vk)? i need help understanding this.

Let V be the vector space of all functions f : R → R. Consider the...

Let V be the vector space of all functions f : R → R. Consider the subspace W spanned by {sin(x), cos(x), e^x , e^−x}. The function T : W → W given by taking the derivative is a linear transformation a) B = {sin(x), cos(x), e^x , e^−x} is a basis for W. Find the matrix for T relative to B. b)Find all the eigenvalues of the matrix you found in the previous part and describe their eigenvectors. (One...

Consider the function and the value of a. f(x) = −2 x − 1 , a...

Consider the function and the value of a. f(x) = −2 x − 1 , a = 9. (a) Use mtan = lim h→0 f(a + h) − f(a) h to find the slope of the tangent line mtan = f '(a). mtan = (b)Find the equation of the tangent line to f at x = a. (Let x be the independent variable and y be the dependent variable.)

1. A vector is parallel to the y-axis. what is its x component? 2. A vector...

1. A vector is parallel to the y-axis. what is its x component? 2. A vector has equal, positive x and y components and zero z component. what is the angle between this vector and the x axis? 3. A vector A has an x component Ax=3 and a y component Ay= -1. what are the x and y components of the vector 2A ? The vector -4A ? 4. What is the magnitude of the vector i + j...

Vector v=(9,0,2) is vector from R3 space. Consider standard inner product in R3. Let W be...

Vector v=(9,0,2) is vector from R3 space. Consider standard inner product in R3. Let W be a subspace in R3 span by u = (9,2,0) and w=(9/2,0,2). a) Does V belong to W? show explanation b) find orthonormal basis in W. Show work c) find projection of v onto W( he best approximation of v with elements of w) d) find the distance between projection and vector v

Consider a continuous random vector (Y, X) with joint probability density function f(x, y) = 1...

Consider a continuous random vector (Y, X) with joint probability density function f(x, y) = 1 for 0 < x < 1, x < y < x + 1. What is the marginal density of X and Y? Use this to compute Var(X) and Var(Y) Compute the expectation E[XY] Use the previous results to compute the correlation Corr (Y, X) Compute the third moment of Y, i.e., E[Y3]

Subjects