praoticemidtermwl.ciIf e was not 0 we could further reduce row I

by subtracting row 3. Hence e -

- O. Also a - o

['

g I § ! bg )c.b.d are free to vary .

because no leading Isthere.

"

l : : :X'

:::H÷÷¥÷÷÷÷÷:÷÷÷:S" l !II! ! ) - I g

'

:3!:/} )Problems at is .

. su .

. i.

If Kit :

fig !! / !NO SOLUTION

⇒ ⇐ :c : ::L !It.÷¥÷÷onElse : -

tooI?" UNFITEnds" ION

2. as In general

8¥two :c:{affection← of L

--etpro ; y

= pro ;¥prog at= cut for same c . we know ut t u

and It= J -

prog so

becauseg

?perpendicular

u . ( T - pros i ) = u . ( v - cut ) = TeoT- Cui. I = O

Hence c=un÷,

so Poser -

-

u÷p

We Know columns of matrix P of projection are Pei,Pez

.

Let

itIII.e- LY. ] .

pros=

uiu=÷µi-fair fun::]pro ;uei=uµ= if :} )

. :P iii.a

iii. I

For 2 a specifically et= [%) ,Huitt 's 25+144=169

: P - talks )

b) Think geometrically

per Fdtlq

Ae ,

→ se, Ace

we see Ae,

= ez and Aez= - e , .

since A =L He, Ak) ,we get A -

-

Lo, I ]

For AZrotateagainrotate again

§Ae '

p nie.

← c-

A ez JAK ,

so A = [to -9 )

c) first note [I ) is NOT on the line.

↳

✓

fu! .

→reft = ref ,

I

refit = projet - it = prog ,J - ( I-prosit ) = 2 pro ; wv

- v

= ( 2 prog ah- Iz )J

So the metro x for the reflection is 2P - Iz where

P is matrix for projection .Hence from Za we get

matrix ref . = Funaki: " "

II. ifFor this specific problem ,

note €1) on line so

refer = I [ I -32 )

→ Indicates on operation was dare

"

l : :÷÷.tt ::÷÷t~ O O 1013¥:it: .

..:)

→ O O 3 → O O I

n *: : :L

" HI title ::'

iii. H : :÷÷. .

)Invertible if K2 - 3kt 2 to

,so k¥1,2

4. write A = LIE ] ,

then A ! I '

y via )

and AAEf.

" Y!! " ) so if AA !Ld 9)

then Hulk I = Hull'

and vi. vz -

- O ( v,

tvz )

i . Solas are all 2×3 matrices with perpendicular

unit vector rows.