LADR Exercises 2A

Problem 1

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

$$\{(x,y,z)\in {\mathbb{F}}^3:x+y+z=0\}$$

We can write $z$ as $-x-y$. Then, we just need linearly independent vectors of the form:

$$(x,y,-x-y)$$

We actually only need two to span the given set, since there are only two variables (this is a plane in 3D space).

Here are four distinct vectors:

$$\begin{array}{r}(1,-1,0)\\ (1,0,-1)\\ (1,1,-2)\\ (-1,1,0)\end{array}$$

Since $x+y+z-0$ is true any of the vectors individually, and any scalar multiplications of the individual vectors, any linear combination of these will always have $x+y+z$ as well.

Problem 2

Prove or give a counterexample: If $v_1,v_2,v_3,v_4$ spans $V$, then the list

$$v_1-v_2,v_2-v_3,v_3-v_4,v_4$$

also spans $V$.

If $v_1,v_2,v_3,v_4$ span $V$, that means any $u\in V$ can be expressed as a linear combination of them. Then, we just need to show that we can produce $v_1,v_2,v_3,v_4$ with the linear combinations four given vectors; if we are able to produce each of these, we will be able to produce everything in the span as well.

$$\begin{array}{rl}{v}_1& =(v_1-v_2)+(v_2-v_3)+(v_3-v_4)+v_4\\ v_2& =(v_2-v_3)+(v_3-v_4)+v_4\\ v_3& =(v_3-v_4)+v_4\\ v_4& =v_4\end{array}$$

Problem 3

Suppose $v_1,\dots ,v_m$ is a list of vectors in $V$. For $k\in \{1,\dots ,m\}$, let

$$w_k=v_1+\cdots +v_k$$

Show that $\text{span}(v_1,\dots ,v_m)=\text{span}(w_1,\dots ,w_m)$.

Problem 4

• (a) Show that a list of length one in a vector space is linearly independent if and only if the vector in the list is not $0$.
• (b) Show that a list of length two in a vector space is linearly independent if and only if neither of the two vectors in the list is a scalar multiple of the other.

Problem 5

Find a number $t$ such that

$$(3,1,4),(2,-3,5),(5,9,t)$$

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

Problem 6

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

Problem 7

• (a) Show that if we think of $\mathbb{C}$ as a vector space over $\mathbb{R}$, then the list $1+i,1-i$ is linearly independent.
• (b) Show that if we think of $\mathbb{C}$ as a vector space over $\mathbb{C}$, then the list $1+i,1-i$ is linearly dependent.

Problem 8

Suppose $v_1,v_2,v_3,v_4$ is linearly independent in $V$. Prove that the list

$$v_1-v_2,v_2-v_3,v_3-v_4,v_4$$

is also linearly independent.

Problem 9

Prove or give a counterexample: If $v_1,v_2,\dots ,v_m$ is a linearly independent list of vectors in $V$, then

$$5v_1-4v_2,v_2,v_3,\dots ,v_m$$

is linearly independent.

Problem 10

Prove or give a counterexample: If $v_1,v_2,\dots ,v_m$ is a linearly independent list of vectors in $V$ and $\lambda \in \mathbb{F}$, then $\lambda v_1,\lambda v_2,\dots ,\lambda v_m$ is linearly independent.

Problem 11

Prove or give a counterexample: If $v_1,\dots ,v_m$ and $w_1,\dots ,w_m$ are linearly independent lists of vectors in $V$, then the list $v_1+w_1,\dots ,v_m+w_m$ is linearly independent.

Problem 12

Suppose $v_1,\dots ,v_m$ is linearly independent in $V$ and $w\in V$. Prove that if $v_1+w,\dots ,v_m+w$ is linearly dependent, then $w\in \text{span}(v_1,\dots ,v_m)$.

Problem 13

Suppose $v_1,\dots ,v_m$ is linearly independent in $V$ and $w\in V$. Show that

$$v_1,\dots ,v_m,w\text{ is linearly independent}⟺w\notin \text{span}(v_1,\dots ,v_m)$$

Problem 14

Suppose $v_1,\dots ,v_m$ is a list of vectors in $V$. For $k\in \{1,\dots ,m\}$, let

$$w_k=v_1+\cdots +v_k$$

Show that the list $v_1,\dots ,v_m$ is linearly independent if and only if the list $w_1,\dots ,w_m$ is linearly independent.

Problem 15

Explain why there does not exist a list of six polynomials that is linearly independent in ${\mathcal{P}}_4(\mathbb{F})$.

Problem 16

Explain why no list of four polynomials spans ${\mathcal{P}}_4(\mathbb{F})$.

Problem 17

Prove that $V$ is infinite-dimensional if and only if there is a sequence $v_1,v_2,\dots$ of vectors in $V$ such that $v_1,\dots ,v_m$ is linearly independent for every positive integer $m$.

Problem 18

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

Problem 19

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

Problem 20

Suppose $p_0,p_1,\dots ,p_m$ are polynomials in ${\mathcal{P}}_m(\mathbb{F})$ such that $p_k(2)=0$ for each $k\in \{0,\dots ,m\}$. Prove that $p_0,p_1,\dots ,p_m$ is not linearly independent in ${\mathcal{P}}_m(\mathbb{F})$.