WeBWorK Standalone Renderer

Statement

Let $V$ be any vector space. Let $r$ be any real number and $\mathbf{v}$ be any vector in $V$ . Prove the following:

Suppose $r \not= 0$ . If $r \, \mathbf{v} = \mathbf{0}$ , then $\mathbf{v} = \mathbf{0}$ .

Suppose $\mathbf{v} \not= \mathbf{0}$ If $r \, \mathbf{v} = \mathbf{0}$ , then $r = 0$ .

Proof

Set the context

Instructions: Put the following statements in order so that they set the context for the proof and define some notation.

Statements to choose from: Drag these statements to the right column.

For full generality, suppose that $V$ is any vector space.
Also, suppose that $\mathbf{v}$ is any vector in $V$ and $r$ is any real number.

Your proof: Put the statements in order in this column and press the Submit Answers button.

Statements to choose from: Drag these statements to the right column.

For full generality, suppose that $V$ is any vector space.
Also, suppose that $\mathbf{v}$ is any vector in $V$ and $r$ is any real number.

Your proof: Put the statements in order in this column and press the Submit Answers button.

Part a

Instructions: Put the following statements in order so that they prove that if $r \not= 0$ and $r \, \mathbf{v} = \mathbf{0}$ , then $\mathbf{v} = \mathbf{0}$ . For uniformity, the proof assumes that you will state what kind of elements you’re working with first, then state what you want to show, then make statements about $r$ , then make statements about $r \, \mathbf{v}$ , and conclude with statements about $\mathbf{v}$ .

Statements to choose from: Drag these statements to the right column.

Show $\mathbf{v} = \mathbf{0}$ .
by substitution and associativity, we have $\mathbf{0} = \frac{1}{r} \mathbf{0}$ $= \frac{1}{r} (r \, \mathbf{v})$ $= (\frac{1}{r} r) \mathbf{v}$ $= 1 \mathbf{v}$ $= \mathbf{v}$ .
We conclude that $\mathbf{v} = \mathbf{0}$ .
$\frac{1}{r}$ exists.
Suppose $r \not= 0$ and $r \, \mathbf{v} = \mathbf{0}$ .
Since $r \, \mathbf{v} = \mathbf{0}$ ,
Since $r \not=0$

Your proof: Put the statements in order in this column and press the Submit Answers button.

Statements to choose from: Drag these statements to the right column.

Show $\mathbf{v} = \mathbf{0}$ .
by substitution and associativity, we have $\mathbf{0} = \frac{1}{r} \mathbf{0}$ $= \frac{1}{r} (r \, \mathbf{v})$ $= (\frac{1}{r} r) \mathbf{v}$ $= 1 \mathbf{v}$ $= \mathbf{v}$ .
We conclude that $\mathbf{v} = \mathbf{0}$ .
$\frac{1}{r}$ exists.
Suppose $r \not= 0$ and $r \, \mathbf{v} = \mathbf{0}$ .
Since $r \, \mathbf{v} = \mathbf{0}$ ,
Since $r \not=0$

Your proof: Put the statements in order in this column and press the Submit Answers button.

Part b

Instructions: Put the following statements in order so that they prove that if $\mathbf{v} \not= \mathbf{0}$ and $r \, \mathbf{v} = \mathbf{0}$ , then $r = 0$ . For uniformity, the proof assumes that you will state what kind of elements you’re working with first, then state what you want to show, then make statements about some possible values of $r$ , and conclude with statements about what value $r$ must have.

Statements to choose from: Drag these statements to the right column.

If $r$ were nonzero, then $r \not= 0$ and $r \, \mathbf{v} = \mathbf{0}$ would imply $\mathbf{v} = \mathbf{0}$ (by part a),
Show that $r = 0$ .
We conclude that $r$ must be $0$ .
but this cannot happen since $\mathbf{v} \not = 0$ .
Suppose that $\mathbf{v} \not= \mathbf{0}$ and $r \, \mathbf{v} = \mathbf{0}$ .

Your proof: Put the statements in order in this column and press the Submit Answers button.

Statements to choose from: Drag these statements to the right column.

If $r$ were nonzero, then $r \not= 0$ and $r \, \mathbf{v} = \mathbf{0}$ would imply $\mathbf{v} = \mathbf{0}$ (by part a),
Show that $r = 0$ .
We conclude that $r$ must be $0$ .
but this cannot happen since $\mathbf{v} \not = 0$ .
Suppose that $\mathbf{v} \not= \mathbf{0}$ and $r \, \mathbf{v} = \mathbf{0}$ .

Your proof: Put the statements in order in this column and press the Submit Answers button.

Advice for more practice

You should use these WebWork proofs as a starting point for learning how to write proofs. This WebWork proof will help you understand the structure of a subspace proof (how statements are ordered). Reviewing several proofs in WebWork and comparing them will help you see similarities and discern a general pattern or outline for proofs. Using these proofs as a template or guide, you should transition to writing your own proofs using a “fill in the blank” approach: “Suppose for full generality that ____ “, “Show that _____ “, “Since _____ has property ___ we know ___ “, “Because of these details _____ we know ____ “, and “Since _____ we conclude that ____ “. Then, deepen your understanding by looking at the details in each step of the WebWork proof and thinking about how you would justify each step and each equality. Finally, after several rounds of practicing proof writing by “filling in the blanks”, try writing a proof from scratch with as little help as possible. Your goal should be to be able to write your own proof from scratch without help. In general, memorizing proofs is not very useful; you’re better off spending your time trying to understand the structural elements that go into writing a proof, understanding the details of a proof, and practicing by writing proofs and getting feedback from others who know how to write proofs.

You can earn partial credit on this problem.