4.1 Exercises

20 points · 4 extra credit

Vector Spaces and Subspaces

1. 1. (2 pts) Determine whether $W = \{(x_1, x_2, x_3) : x_1 - 3x_2 + x_3 = 0\}$ is a subspace of $\mathbb{R}^3$. Justify your answer using the subspace test.

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2. 2. (2 pts) Determine whether $W = \{(x_1, x_2) : x_1 x_2 \ge 0\}$ (the union of the first and third quadrants) is a subspace of $\mathbb{R}^2$. Justify your answer.

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3. 3. (3 pts) Let $H$ be the set of all $2\times 2$ matrices $A$ satisfying $A^{\mathsf T} = A$ (the symmetric matrices). Show that $H$ is a subspace of the vector space $M_{2\times 2}$.

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4. 4. (3 pts) Work in $P_2$, the space of polynomials of degree at most $2$.

   (a) Show that $U = \{p : p(1) = 0\}$ is a subspace of $P_2$.

   (b) Show that $\{p : p(1) = 2\}$ is not a subspace of $P_2$.

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5. 5. (2 pts) The plane $2x - y + 3z = 0$ in $\mathbb{R}^3$ passes through the origin. Write it as $\text{Span}\,\{\mathbf{u}, \mathbf{v}\}$ for two explicit vectors, thereby exhibiting it as a subspace.

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6. 6. (3 pts) State whether each of the following is true or false, and justify each answer.

   (a) The set $\{\mathbf{0}\}$ is a subspace of every vector space.

   (b) If $\mathbf{u}$ and $\mathbf{v}$ lie in a subspace $H$, then $3\mathbf{u} - 5\mathbf{v}$ lies in $H$.

   (c) Every line in $\mathbb{R}^2$ is a subspace of $\mathbb{R}^2$.

   (d) The empty set is a subspace of every vector space.

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7. 7. (3 pts) Let $H$ and $K$ be subspaces of a vector space $V$. Prove that the intersection $H \cap K$ is a subspace of $V$.

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8. 8. (2 pts) Give an explicit example of two subspaces $H$ and $K$ of $\mathbb{R}^2$ whose union $H \cup K$ is not a subspace of $\mathbb{R}^2$.

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9. 9. (4 pts ••) Let $V$ be a vector space. Using only the axioms, prove that the zero vector is unique and that each $\mathbf{v} \in V$ has a unique additive inverse.

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