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Vector calculus: exam 1

1. Represent vector $(1,1)$ as a linear combination of vectors $(1,2)$ and $(-1,3)$.
3. For the linear equation $$x_1+2x_2+x_3=0$$ compute two vectors $u$ and $v$ such that every solution $x=(x_1,x_2,x_3)$ can be written as a linear combination of $u$ and $v$.
4. Solve the following system, expressing the solution in vector form: $$x_2-x_3=1$$ $$x_1+x_2+x_3=2.$$
5. Let $S={\rm span \hspace{3pt}} \{v \}$, where $v=(2,-1)$. (a) Compute the homogeneous linear equation in $x_1,x_2$ that determines $S$. (b) Suppose $A$ is an affine subspace parallel to $S$ and passing through $w=(3,0)$. Compute the linear equation that determines $A$.
6. Prove that any four vectors in $R^3$ are linearly dependent. Use the definition of linear independence and properties of $R^3$.
7. Prove the homogeneity property of $||x||$.
8. Find the projection of $v=(1,2,3)$ onto the plane spanned by $(1,1,1)$ and $(0,1,2)$.
9. Find all unit vectors perpendicular to the plane spanned by $(-1,1,0)$ and $(0,1,1)$.