Consider the R-vector space V-R2. Which of the following subsets of V is a subspace? The line 2x+y=1 O The line y = 2x O The circle x²+y²=1 O The set {(x, y) | x ≥ 0 and y ≥ 0}

Prove that in a given vector space V, the zero vector is unique.

Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector u=(1,1,1,1) in the form u=v+w, where v is in V and w is orthogonal to every vector in V.

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].

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.

Consider the vectors u=(6,2,4) and v=(1,2,0) from Example 10. Without using Theorem 5.9, show that among all the scalar multiples cv of the vector v, the projection of u onto v is the closest to u that is, show that d(u,projvu) is a minimum.

Prove that in a given vector space V, the additive inverse of a vector is unique.

In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=3, W={[a0a]}

Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).

In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=3, W={[aba]}