Definition: Let V be a vector space and U₁, U₂, ,Uk be subspaces of V. Then V is said to be a direct sum of subspaces U₁, U2,...,Uk, denoted by, V = U₁ ĐU₂ ☺…ÐUk, if the following two conditions hold: V=U₁+U₂+... + Uk; (ii) For every v € V, there exist unique vectors uį € Uį, 1 ≤ i ≤ k, such that V = U₁ + ··· + Uk. 9. (a) Suppose that U₁,...,Uk are subspaces of V. Prove that V = U₁ ··· Uk if and only if the following two conditions hold: (i) V=U₁+...+Uk. Proved. (ii) The only way to write Oy as a sum of u₁ + ··· + uk, where each u; € Uj, is by taking all u,'s equal to Proved. zero. (b) Suppose that V is a finite dimensional vector space, with dim(V) = n. Prove that there exist 1-dimensional subspaces U₁,..., Un of V such that V = U₁₁ U₁₂ Un. Proved. Give an example to show that condition (ii) (in definition) can not be replaced with U₂ nu; = {0v}, for i ‡ j.

Let v1=[-2,2,1], v2=[-8,7,4], v3=[-29,25,14], w=[6,6,1] is W in {v1,v2,v3}? how many vectors are in {v1,v2,v3}? How many vectors are in span {v1,v2,v3}? is w in the subspace spanned by {v1,v2,v3}?

Can you explain why the statement in the image is true?

28. Is C a real vector space? Explain.