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Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CM: Cumulative Review

Problem 24CM

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Let V and V' be finite dimensional
rector spaces over a field
I and t: V--V' be
a linear
transformation and to be the dual
map
of t, then t and t* have the
Same rank.

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Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1) is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).
3. Show that in a normed space X, vector addition and multiplication by scalars are continuous operations with respect to the norm.
Let Rz, be a linear space with all polynomials p(z) = ax² + bx+c(a,b,c € R) whose degree is no more than 2. Let L be the operator on R[r], defined by 4 Lar²+ bx+c)=(a + 3b+6c)x²+(-a-3b-4c)x+(a+b+ 3 (1) Find the matrix A representing with respect to basis [1, 2, 3x²]; 2 3 (2) Find the matrix B representing with respect to basis (1-2r+32²,-1+ 2r, -1+3z²); (3) Is the matrix A diagonalizable? Why?
Let P2 be the vector space of all real polynomials of degree at most 2. Let p1, P2, P3 E P2 be given by p1(x) = 2, p2(x) = x + 2x², and p3(x) = 6x + ax². а) (You are free to assume that the functions 1, x and x? are linearly independent.) b) combination of p1, P2 and p3. %3D Find the condition on a e R that ensures that {P1, P2, P3} is a basis for P2. In the case that a = 6, write the function p(x) = 2 – x + x² as a linear %D
5. Let the set V be given as the set of all vectors in R^2 , except the zero vector. Define a relation R on V as followsu R v if and only if u · v 0 where “·” denotes the scalar product, ie. (x1, y1) · (x2, y2)= x1x2 + y1y2 for all(x1, y1), (x2, y2) ∈ R ^2. Determine whether R is an equivalence relation.
Let P3 is the vector space of polynomials of the degree<3,Consider the polynomials P1 t)=1+t,^p2(t)=1-t, is{p1, P2} a linearly independent set in P3? Why or why not.
Let V, W be inner-product spaces.
1) Let V be the set of all ordered pairs (x, y) of real numbers and F be the field of real numbers. Define (x, y) + (X1, y2) = (3y + 3 yı, -x - x1) and c[x, y) = (3cy, -cx). Verify that V, with these operations, is not a vector space over the field of real numbers.
Let X be a vector space. Let || -| and ||-|| be two norms on X. When are these norms said to be equivalent? Justify your answer. Let X = R³. For X = (X₁, X₂, X3). Let ||x|| ' = x₁ |+|x₂|+|X3| ||1x||² Show that = √1x₁ 1² + 1x₂ 1² + 1x3|² X1 and | || are equivalent.
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).

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