Dimension

Show that the subspaces of $\mathbb {R}^{2}$ are precisely $\left\{ 0\right\}$, all lines in $\mathbb {R}^{2}$ containing the origin, and $\mathbb {R}^{2}$.

Show that the subspaces of $\mathbb {R}^{3}$ are precisely $\left\{ 0\right\}$, all lines in $\mathbb {R}^{3}$ containing the origin, all planes in $\mathbb {R}^{3}$ containing the origin, and $\mathbb {R}^{3}$.

Suppose $v_{1}, \ldots , v_{m}$ is linearly independent in $V$ and $w \in V$. Prove that

Suppose $m$ is a positive integer and $p_{0}, p_{1}, \ldots , p_{m} \in \mathcal{P}(\mathbf{F})$ are such that each $p_{k}$ has degree $k$. Prove that $p_{0}, p_{1}, \ldots , p_{m}$ is a basis of $\mathcal{P}_{m}(\mathbf{F})$.

Suppose $m$ is a positive integer. For $0 \leq k \leq m$, let

Show that $p_{0}, \ldots , p_{m}$ is a basis of $\mathcal{P}_{m}(\mathbf{F})$. The basis in this exercise leads to what are called Bernstein polynomials. You can do a web search to learn how Bernstein polynomials are used to approximate continuous functions on $[0,1]$.

Suppose $U$ and $W$ are both four-dimensional subspaces of $\mathbb {C}^{6}$. Prove that there exist two vectors in $U \cap W$ such that neither of these vectors is a scalar multiple of the other.

Suppose that $U$ and $W$ are subspaces of $\mathbb {R}^{8}$ such that $\operatorname {dim} U = 3, \operatorname {dim} W = 5$, and $U+W = \mathbb {R}^{8}$. Prove that $\mathbb {R}^{8} = U \oplus W$.

Suppose $U$ and $W$ are both five-dimensional subspaces of $\mathbb {R}^{9}$. Prove that $U \cap W \neq \left\{ 0\right\}$.

Suppose $V$ is a ten-dimensional vector space and $V_{1}, V_{2}, V_{3}$ are subspaces of $V$ with $\operatorname {dim} V_{1} = \operatorname {dim} V_{2} = \operatorname {dim} V_{3} = 7$. Prove that $V_{1} \cap V_{2} \cap V_{3} \neq \left\{ 0\right\}$.

Suppose $V$ is finite-dimensional and $V_{1}, V_{2}, V_{3}$ are subspaces of $V$ with $\operatorname {dim} V_{1}+\operatorname {dim} V_{2}+\operatorname {dim} V_{3} > 2 \operatorname {dim} V$. Prove that $V_{1} \cap V_{2} \cap V_{3} \neq \left\{ 0\right\}$.

Suppose $V$ is finite-dimensional and $U$ is a subspace of $V$ with $U \neq V$. Let $n = \operatorname {dim} V$ and $m = \operatorname {dim} U$. Prove that there exist $n-m$ subspaces of $V$, each of dimension $n-1$, whose intersection equals $U$.

Suppose that $V_{1}, \ldots , V_{m}$ are finite-dimensional subspaces of $V$. Prove that $V_{1}+\cdots +V_{m}$ is finite-dimensional and

The inequality above is an equality if and only if $V_{1}+\cdots +V_{m}$ is a direct sum, as will be shown in 3.94.

Suppose $V$ is finite-dimensional, with $\operatorname {dim} V = n \geq 1$. Prove that there exist one-dimensional subspaces $V_{1}, \ldots , V_{n}$ of $V$ such that

Explain why you might guess, motivated by analogy with the formula for the number of elements in the union of three finite sets, that if $V_{1}, V_{2}, V_{3}$ are subspaces of a finite-dimensional vector space, then

Then either prove the formula above or give a counterexample.

Prove that if $V_{1}, V_{2}$, and $V_{3}$ are subspaces of a finite-dimensional vector space, then

The formula above may seem strange because the right side does not look like an integer.