1. (i) Give the definition of a norm on a vector space X. [5 Marks] (ii) Explain why the function ||x|| = |x1 + x2| does not define a norm on the vector space R². [15 Marks]
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![1. (i) Give the definition of a norm on a vector space X.
[5 Marks]
(ii) Explain why the function ||x|| = |x1 + x2| does not define a norm on
the vector space R².
[15 Marks]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39a0c022-e49a-477b-ab1a-da6fafebfbe8%2F45fc30b3-81ac-406f-a41e-62604ad11c40%2F713dz5j_processed.jpeg&w=3840&q=75)
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- Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}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.Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.
- Let V be the set of all positive real numbers. Determine whether V is a vector space with the operations shown below. x+y=xyAddition cx=xcScalar multiplication If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].Let v1, v2, and v3 be three linearly independent vectors in a vector space V. Is the set {v12v2,2v23v3,3v3v1} linearly dependent or linearly independent? Explain.
- 3. [6 points] Consider the vector space V = P2 over the field of real numbers F = R. Is the vector 4 – x – x² in the span of the vectors 1+ 2x, x – x², -1+x+x² Explain why or why not. Show your calculations.Let v1 and v2 be vectors in an inner product space V.(a) Is it possible for |{v1, v2} | to be greater than||v1|| ||v2||? Explain.(b) If| {v1, v2}| = ||v1|| ||v2||what can you conclude about the vectors v1 andv2? Explain.#Consider the number line R. Let a "vector" be a real number and the scalars a & b in properties (e), (f), (g) on pg 223 can only be rational numbers. Show that R is a vector space over the rational numbers Q, using standard addition and multiplication of real numbers.
- Show that the function f: R² → R defined by f(x₁, x₂) = x² + 2x² + 3x₁x₂ + 4x₁ + 5x₂ + 6 is a vector quadratic function. HINT: Start by saying how 'vector quadratic function' is defined.1 (a) Is the vector| 2 in the span of -1 and 2 ? Justify your answer 3 3 2 2 (b) Is the set linearly independent or linearly dependent? Justify your answer.Which of the following are vector spaces? Justify your answer. (a) The set of all polynomials of degree 3. (b) The set of all vectors x = (x1, x2, x3), satisfying 3x₁ + 5x2 − 9x3 = 2023. (c) The set of all vectors x = (x1, x2, x3), satisfying 2024x₁ + x2 = 0 has a unique solution. (d) The set of all 3 × 3 matrices such that Ax= (e) The set of all n × n (n = N) diagonal matrices. - x3 = 0. -
