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.What is the dimension of the complexification VC thought of as a real vector space?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.
