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# Vector calculus: exam 1

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Jump to navigationJump to searchThese are exercises for Vector calculus: course.

- Represent vector $(1,1)$ as a linear combination of vectors $(1,2)$ and $(-1,3)$.
- Is the union of two linear subspaces always a linear subspace? Prove or provide a counterexample.
- 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 $v$.
- Solve the following system, expressing the solution in vector form: $$x_2-x_3=1$$ $$x_1+x_2+x_3=2.$$
- 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 $A$.
- Prove that any four vectors in $R^3$ are linearly dependent. Use the definition of linear independence and properties of $R^3$.
- Prove the homogeneity property of $||x||$.
- Find the projection of $v=(1,2,3)$ onto the plane spanned by $(1,1,1)$ and $(0,1,2)$.
- Find all unit vectors perpendicular to the plane spanned by $(-1,1,0)$ and $(0,1,1)$.