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]

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.

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.

one example of a 4-dimensional vector space that is not R 4

Confused. Please help:)

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.