Finite-Dimensional Vector Spaces

Find a list of four distinct vectors in $\mathbf{F}^{3}$ whose span equals

Prove or give a counterexample: If $v_{1}, v_{2}, v_{3}, v_{4}$ spans $V$, then the list

also spans $V$.

Suppose $v_{1}, \ldots , v_{m}$ is a list of vectors in $V$. For $k \in \left\{ 1, \ldots , m\right\}$, let

Show that $\operatorname {span}\left(v_{1}, \ldots , v_{m}\right) = \operatorname {span}\left(w_{1}, \ldots , w_{m}\right)$.

Find a number $t$ such that

is not linearly independent in $\mathbb {R}^{3}$.

Show that the list (2,3,1), (1, -1,2), (7,3,c) is linearly dependent in $\mathbf{F}^{3}$ if and only if $c = 8$.

Suppose $v_{1}, v_{2}, v_{3}, v_{4}$ is linearly independent in $V$. Prove that the list

is also linearly independent.

Prove or give a counterexample: If $v_{1}, v_{2}, \ldots , v_{m}$ is a linearly independent list of vectors in $V$, then

is linearly independent.

Prove or give a counterexample: If $v_{1}, v_{2}, \ldots , v_{m}$ is a linearly independent list of vectors in $V$ and $\lambda \in \mathbf{F}$ with $\lambda \neq 0$, then $\lambda v_{1}, \lambda v_{2}, \ldots , \lambda v_{m}$ is linearly independent.

Prove or give a counterexample: If $v_{1}, \ldots , v_{m}$ and $w_{1}, \ldots , w_{m}$ are linearly independent lists of vectors in $V$, then the list $v_{1}+w_{1}, \ldots , v_{m}+w_{m}$ is linearly independent.

Suppose $v_{1}, \ldots , v_{m}$ is linearly independent in $V$ and $w \in V$. Prove that if $v_{1}+w, \ldots , v_{m}+w$ is linearly dependent, then $w \in \operatorname {span}\left(v_{1}, \ldots , v_{m}\right)$.

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

Suppose $v_{1}, \ldots , v_{m}$ is a list of vectors in $V$. For $k \in \left\{ 1, \ldots , m\right\}$, let

Show that the list $v_{1}, \ldots , v_{m}$ is linearly independent if and only if the list $w_{1}, \ldots , w_{m}$ is linearly independent.

Explain why there does not exist a list of six polynomials that is linearly independent in $\mathcal{P}_{4}(\mathbf{F})$.

Explain why no list of four polynomials spans $\mathcal{P}_{4}(\mathbf{F})$.

Prove that $V$ is infinite-dimensional if and only if there is a sequence $v_{1}, v_{2}, \ldots$ of vectors in $V$ such that $v_{1}, \ldots , v_{m}$ is linearly independent for every positive integer $m$.

Prove that $\mathbf{F}^{\infty }$ is infinite-dimensional.

Prove that the real vector space of all continuous real-valued functions on the interval $[0,1]$ is infinite-dimensional.

Suppose $p_{0}, p_{1}, \ldots , p_{m}$ are polynomials in $\mathcal{P}_{m}(\mathbf{F})$ such that $p_{k}(2) = 0$ for each $k \in \left\{ 0, \ldots , m\right\}$. Prove that $p_{0}, p_{1}, \ldots , p_{m}$ is not linearly independent in $\mathcal{P}_{m}(\mathbf{F})$.

Find all vector spaces that have exactly one basis.

Verify all assertions in Example 2.27.

Example 2.27:

Suppose $V$ is finite-dimensional and $U, W$ are subspaces of $V$ such that $V = U+W$. Prove that there exists a basis of $V$ consisting of vectors in $U \cup W$.

Prove or give a counterexample: If $p_{0}, p_{1}, p_{2}, p_{3}$ is a list in $\mathcal{P}_{3}(\mathbf{F})$ such that none of the polynomials $p_{0}, p_{1}, p_{2}, p_{3}$ has degree 2, then $p_{0}, p_{1}, p_{2}, p_{3}$ is not a basis of $\mathcal{P}_{3}(\mathbf{F})$.

Suppose $v_{1}, v_{2}, v_{3}, v_{4}$ is a basis of $V$. Prove that

is also a basis of $V$.

Prove or give a counterexample: If $v_{1}, v_{2}, v_{3}, v_{4}$ is a basis of $V$ and $U$ is a subspace of $V$ such that $v_{1}, v_{2} \in U$ and $v_{3} \notin U$ and $v_{4} \notin U$, then $v_{1}, v_{2}$ is a basis of $U$.

Suppose $v_{1}, \ldots , v_{m}$ is a list of vectors in $V$. For $k \in \left\{ 1, \ldots , m\right\}$, let

Show that $v_{1}, \ldots , v_{m}$ is a basis of $V$ if and only if $w_{1}, \ldots , w_{m}$ is a basis of $V$.

Suppose $U$ and $W$ are subspaces of $V$ such that $V = U \oplus W$. Suppose also that $u_{1}, \ldots , u_{m}$ is a basis of $U$ and $w_{1}, \ldots , w_{n}$ is a basis of $W$. Prove that

is a basis of $V$.

Suppose $V$ is a real vector space. Show that if $v_{1}, \ldots , v_{n}$ is a basis of $V$ (as a real vector space), then $v_{1}, \ldots , v_{n}$ is also a basis of the complexification $V_{\mathrm{C}}$ (as a complex vector space).

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.