What is Linear Algebra?

€

ry = A

r x +

r b

Notation:

Linear Transformation

Linear OperatorMatrix Multiplication

n-Dimensional Linear Mapping

€

ry = A

r x

Linear Coordinate Transformation

Matrix Multiplication

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ry = A

r x

7

6

3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

0 1 01 0 11 −1 3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

472

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Just a common kind of function

Mappings from Input to Outputs

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y = f (x)

€

x

€

y

€

fdoseresponse

contrast firing ratecolumn

space

row space

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ry = A

r x

Linear Transforms:x1

x2a

a

Linear Transforms:Identity

b

Linear Transforms:Stretch and Squash

ab

Linear Transforms:Flips

b

a

Linear Transforms:Rotation

b

a

Linear Transforms:Skew

ab

Linear Transforms:Any Combination of the Above

b

a

Linear Transforms:

€

RFSRSSr x

€

RFSRSS = A

Mix and Match, Collect the Whole Set

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=Ar x

Singular Value Decomposition:

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A = UDVAny Linear Transform can be written as:

a rotation, a stretch and flip, and another rotation

EigenVectors:

Vat gözinta, cümzouta.

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λ r

e = Ar e

‘eigen’ is German for ‘self’

a

Linear Transforms:Identity

b

Linear Transforms:Flips

b

a

Linear Transforms:Rotation

b

a

But What’s this good for?

fMRI

Time series realignment

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Brainrealigned = M ⋅Brainmisaligned

3D Rigid-body Transformations• A 3D rigid body transform is defined by:

– 3 translations - in X, Y & Z directions

– 3 rotations - about X, Y & Z axes

• The order of the operations matters

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1 0 0 Xtrans

0 1 0 Ytrans

0 0 1 Ztrans

0 0 0 1

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

×

1 0 0 0

0 cosθ sinθ 0

0 −sinθ cosθ 0

0 0 0 1

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

×

cosφ 0 sinφ 0

0 1 0 0

−sinφ 0 cosφ 0

0 0 0 1

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

×

cosψ sinψ 0 0

−sinψ cosψ 0 0

0 0 1 0

0 0 0 1

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

Translations Pitchabout x axis

Rollabout y axis

Yawabout z axis

Linear Regression

Linear Regression

Linear Regression

€

ry obs

€

rx

Linear Regression

€

ry est = β

r x

€

min E = (r y est −

r y obs)

2

Linear Regression

€

ry obs

€

rx

€

ry est = β

r x

€

min E = (r y est −

r y obs)

2

Linear Regression

€

Y = Xr β

€

ry = β 0 + β1

r x 1 + β 2

r x 2

Linear Regression

€

Y = Xr β

€

rβ =(X ' X)−1 X 'Y

fMRI

Intensity

Tim

e

Regression model

= β1 β2+ + erro

r

x1 x2 ε

ε∼Ν0 σ2Ι(error is normal andindependently and

identically distributed)

Question: Is there a change in the BOLD response between listening and rest?

Question: Is there a change in the BOLD response between listening and rest?

Hypothesis test: β1 = 0?(using t-statistic)

General case

Y = β1 X1 β2 X2 ε

(1,1,1)

(x1, x2, x3)X1

X2O

Y1 x1 1 β1 ε1

Y2 = x2 1 + ε2

Y3 x3 1 β2 ε3

DATA(Y1, Y2, Y3)

Y

design space

Geometrical perspective

Linear Regression

€

ry obs

€

rx

What is Differential Equation?

€

d

dtf t( ) = g t( )

Let’s take this a bit further:

What is Differential Equation?

€

d

dt

r f t( ) =

r g t( )

Let’s take this a bit further:

€

ry = A

r x

And you all remember linear systems:

Linear Dynamical Systems!

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d

dt

r f t( ) = A

r f t( )

Can you read this?

Phase Space:f1

f2

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rf 0( )

€

d

dt

r f t( ) = A

r f t( )

What happens if Ais a stretcher?

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rf 1( )

€

rf 2( )

Differential Linear Systems

€

d

dt

r f t( ) = A

r f t( )

It is the nature of A which determines the behavior of the system

Differential Linear Systems

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d

dt

r f t( ) = A

r f t( )

The return of Eigenvectors...

Differential Linear Systems

Nonlinear Dynamical Systems