Linear Algebra Exam 1 Chapter 1 Defintions

Yeah — that study method is much better than trying to grind through dozens of textbook problems tonight. And there's one important reason: Bennett has already told you the exam is going to mix definitions, conceptual understanding, and computations. So we should train those separately and then combine them in a mock exam.

Your plan should be:

Definitions→Concepts→Section problems→Practice Exam\boxed{\text{Definitions}\rightarrow\text{Concepts}\rightarrow\text{Section problems}\rightarrow\text{Practice Exam}}

And for the problems, I like your idea of not immediately showing you the solution. We'll use a hint ladder. You attempt it → if stuck, I tell you the first move → you continue → if still stuck, I give the next move → only after that do we fully work it. That forces you to actually learn how to recognize what to do.

Bennett also said the review-guide checklist is essentially the checklist for being fully prepared for Wednesday, so we don't need to invent extra material tonight.

Phase 1 starts now: the 7 definitions

Don't try memorizing my explanation. Memorize the bold definition, then use the short phrase underneath to recall it.

1. Span — definitely know this

Span⁡{v1,…,vp} is the set of all linear combinations of v1,…,vp.\boxed{ \operatorname{Span}\{\mathbf v_1,\ldots,\mathbf v_p\} \text{ is the set of all linear combinations of } \mathbf v_1,\ldots,\mathbf v_p. }

Expanded:

Span⁡{v1,…,vp}={c1v1+⋯+cpvp:c1,…,cp∈R}.\operatorname{Span}\{\mathbf v_1,\ldots,\mathbf v_p\} = \{c_1\mathbf v_1+\cdots+c_p\mathbf v_p: c_1,\ldots,c_p\in\mathbb R\}.

Memory: Span = ALL linear combinations.

And Bennett specifically told the class that Span will be one of the definitions appearing on the exam.

2. Linear combination

y is a linear combination of v1,…,vp if y=c1v1+⋯+cpvp for some scalars c1,…,cp.\boxed{ \mathbf y\text{ is a linear combination of } \mathbf v_1,\ldots,\mathbf v_p \text{ if } \mathbf y=c_1\mathbf v_1+\cdots+c_p\mathbf v_p \text{ for some scalars }c_1,\ldots,c_p. }

Memory: multiply vectors by weights, then add.

Notice the difference:

linear combination=ONE resulting vector\text{linear combination}=\textbf{ONE resulting vector}span=ALL possible resulting vectors.\text{span}=\textbf{ALL possible resulting vectors}.

That's a very testable conceptual distinction.

3. Linearly independent

{v1,…,vp} is linearly independent if c1v1+⋯+cpvp=0 has only the trivial solution.\boxed{ \{\mathbf v_1,\ldots,\mathbf v_p\} \text{ is linearly independent if } c_1\mathbf v_1+\cdots+c_p\mathbf v_p=\mathbf0 \text{ has only the trivial solution.} }

Meaning:

c1=c2=⋯=cp=0.\boxed{c_1=c_2=\cdots=c_p=0}.

Memory: Independent = ONLY ZERO weights work.

Bennett directly connected this definition to Ax=0A\mathbf x=\mathbf0 during review.

4. Linearly dependent

{v1,…,vp} is linearly dependent if there exist scalars c1,…,cp, not all zero, such that c1v1+⋯+cpvp=0.\boxed{ \{\mathbf v_1,\ldots,\mathbf v_p\} \text{ is linearly dependent if there exist scalars } c_1,\ldots,c_p,\text{ not all zero, such that } c_1\mathbf v_1+\cdots+c_p\mathbf v_p=\mathbf0. }

That's a nontrivial solution.

Memory: Dependent = NONZERO weight possible.

So never mix these up:

Independent  ⟺  only trivial solution\boxed{\text{Independent}\iff\text{only trivial solution}}Dependent  ⟺  nontrivial solution exists\boxed{\text{Dependent}\iff\text{nontrivial solution exists}}

5. Linear transformation

T:Rm→Rn is linear if, for all u,v and scalars c,\boxed{ T:\mathbb R^m\rightarrow\mathbb R^n \text{ is linear if, for all }\mathbf u,\mathbf v \text{ and scalars }c, }T(u+v)=T(u)+T(v)\boxed{T(\mathbf u+\mathbf v)=T(\mathbf u)+T(\mathbf v)}

and

T(cu)=cT(u).\boxed{T(c\mathbf u)=cT(\mathbf u)}.

Both conditions are required.

Memory: linear transformations preserve addition AND scalar multiplication.

A useful consequence:

T(0)=0.\boxed{T(\mathbf0)=\mathbf0}.

So something like

T(x1,x2)=(x1,x2+5)T(x_1,x_2)=(x_1,x_2+5)

can't be linear because

T(0,0)=(0,5)≠(0,0).T(0,0)=(0,5)\neq(0,0).

6. One-to-one

T is one-to-one if each vector in the range is the image of at most one vector in the domain.\boxed{ T\text{ is one-to-one if each vector in the range is the image of at most one vector in the domain.} }

Bennett described this as a uniqueness question.

Connect the definition to:

T one-to-one  ⟺  Ax=0 has only the trivial solution  ⟺  pivot in every column  ⟺  columns independent.\boxed{ T\text{ one-to-one} \iff A\mathbf x=\mathbf0\text{ has only the trivial solution} \iff \text{pivot in every column} \iff \text{columns independent}. }

Memory: ONE-to-one → one unique input → every COLUMN pivots.

7. Onto

T:Rm→Rn is onto if every vector in the codomain is the image of at least one vector in the domain.\boxed{ T:\mathbb R^m\rightarrow\mathbb R^n \text{ is onto if every vector in the codomain is the image of at least one vector in the domain.} }

Equivalently:

Range⁡(T)=Codomain⁡(T).\boxed{\operatorname{Range}(T)=\operatorname{Codomain}(T)}.

Bennett described this as an existence question.

Connect it to:

T onto  ⟺  pivot in every row  ⟺  columns of A span the codomain.\boxed{ T\text{ onto} \iff \text{pivot in every row} \iff \text{columns of }A\text{ span the codomain}. }

Memory: Onto → covers the whole output space → every ROW pivots.

Your first memorization test

Don't move to concepts yet.

Read those seven, then hide everything above this point and, in your next message, try to give me the definitions of these from memory:

Span → Linear combination → Linearly independent → Linearly dependent → Linear transformation → One-to-one → Onto

They don't need to be beautifully worded on your first attempt. Write what you genuinely remember.

I'll grade each one as ✓ exam-ready, △ basically right but needs fixing, or ✗ relearn, and we'll repeat only the ones you miss.

Once all seven are ✓, then we'll attack the working concepts: RnR^n, domain/codomain/range, pivots, basic/free variables, trivial/nontrivial solutions, homogeneous/nonhomogeneous, REF/RREF, standard basis, standard matrix, span/independence theorems, dimension shortcuts, and the rotation matrix. Then we start 1.1 and work straight through to 1.9 with the hint system you described.

That's a pretty solid way to turn the energy you have right now into actual recall instead of just rereading notes.

For these definitions, don't memorize them by rereading them 20 times. You want to train yourself to produce them, because that's what Bennett is making you do on the exam.

I'd use a keyword → skeleton → exact definition method.

Step 1: Memorize just these 7 tiny triggers

Get these seven down first. They're much easier to hold in your head than seven paragraphs.

Step 2: Attach the mathematics to each trigger

For example, say out loud:

Independent → only trivial solution →

c1v1+⋯+cpvp=0c_1v_1+\cdots+c_pv_p=0

has only

c1=⋯=cp=0.c_1=\cdots=c_p=0.

Then:

Dependent → nontrivial solution →

c1v1+⋯+cpvp=0c_1v_1+\cdots+c_pv_p=0

has a solution where the cic_i's are not all zero.

Those two should be learned together because they're opposites.

Do the same with one-to-one and onto:

One-to-one = AT MOST one.
Each vector in the codomain is the image of at most one vector in the domain.

Onto = AT LEAST one.
Each vector in the codomain is the image of at least one vector in the domain.

That MOST vs LEAST distinction is a really nice memory hook.

Step 3: Use active recall

Grab a blank sheet. Write:

1. Span =

2. Linear combination =

3. Independent =

4. Dependent =

5. Linear transformation =

6. One-to-one =

7. Onto =

Then close your notes.

Try filling all seven from memory. Don't look when you get stuck for five seconds. Struggle with it for maybe 15–20 seconds first—that struggle is actually useful for memorization.

Then check your answers.

Put a ✓ beside correct ones and circle anything you forgot.

Only redo the circled ones.

Step 4: Do three rounds

Your first round might look like:

Span = all linear combinations.

Good. Now expand it:

Span of v1,…,vpv_1,\ldots,v_p is the set of all linear combinations of v1,…,vpv_1,\ldots,v_p.

Then eventually you should be able to write:

Span⁡{v1,…,vp} is the set of all linear combinations of v1,…,vp.\boxed{ \operatorname{Span}\{v_1,\ldots,v_p\} \text{ is the set of all linear combinations of } v_1,\ldots,v_p. }

That's exam-ready.

Do the same three-round process with every definition:

Round 1: remember the trigger.
Round 2: say the definition aloud.
Round 3: write it without looking.

Step 5: Mix the order

This part matters.

Don't always memorize:

Span → Combination → Independent → Dependent...

Otherwise your brain can accidentally memorize the sequence instead of the definitions.

After you've learned them, shuffle:

Define onto.

Then:

Define linear combination.

Then:

Define linearly dependent.

Then:

Define span.

That's much closer to Bennett randomly choosing 3–5 definitions for the first page. She specifically said the exam definitions will be a subset of the review-guide list, with Span and either dependence or independence among them.

One little chant that covers almost everything

I'd actually memorize this:

Span — ALL combinations.
Combination — weighted sum.
Independent — ONLY trivial.
Dependent — NONTRIVIAL.
Linear — preserves addition and scaling.
One-to-one — AT MOST one.
Onto — AT LEAST one for EVERY output.

Once you can say that without hesitation, expanding each into Bennett's full mathematical definition becomes way easier.

And then connect the concepts:

Independent↔pivot every COLUMN↔one-to-one\boxed{\text{Independent}\leftrightarrow\text{pivot every COLUMN}\leftrightarrow\text{one-to-one}}Span codomain↔pivot every ROW↔onto\boxed{\text{Span codomain}\leftrightarrow\text{pivot every ROW}\leftrightarrow\text{onto}}

Don't try to memorize definitions + every theorem + every section simultaneously. Get these seven locked first. Then we can immediately quiz you on them one at a time—without showing you the answers—until you can produce all seven cold.