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All of Linear Algebra

in Five Minutes

Aaron Schumacher

Thursday 2026-10-15

Data Science DC

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month

flowers

3

6

5

10

7

14

12

​

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month

flowers

3

6

5

10

7

14

12

24

month

flowers

3

6

5

10

7

14

5 + 7

10 + 14

“additive”

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month

flowers

3

6

5

10

7

14

12

24

month

flowers

3

6

5

10

7

14

4 × 3

4 × 6

“homogeneous”

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month

flowers

3

6

5

10

7

14

12

24

month

flowers

3

2 × 3

5

2 × 5

7

2 × 7

12

2 × 12

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f(x) = ax

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Additive:

f(a + b) = f(a) + f(b)

​

Homogeneous:

f(c × a) = c × f(a)

​

​

​

​

f(x) = x × f(1)

Choose f(1) = a

f(x) = ax

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f(x) = ax

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0

0

0.5

1

1

2

1.5

3

2

4

→

× 2

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0

0

0.5

1

1

2

1.5

3

2

4

→

× 2

←

× 0.5

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0

0

0.5

0.5

1

1

1.5

1.5

2

2

→

× 1

←

× 1

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0

0

0.5

0

1

0

1.5

0

2

0

→

× 0

←

😱

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-2

4

-1

2

0

0

1

-2

2

-4

→

× -2

←

× -0.5

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-2 = 2 × -1

length

orientation

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a × (2 × -1)

►► +1 (cosign)

◄◄ +1 (cosign)

◄► -1 (disagree)

►◄ -1 (disagree)

​

0

0

1

2

2

4

3

6

4

8

lengths

orientations

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-2

-2

-1

-1

0

0

1

1

2

2

-2

2

-1

1

0

0

1

-1

2

-2

× 1

× -1

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f(x, y) = ax + by

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Additive:

f(a + b) = f(a) + f(b)

​

Homogeneous:

f(c × a) = c × f(a)

​

f(x, y)

= f(x, 0) + f(0, y)

​

= x × f(1, 0) + y × f(0, 1)

Choose f(1, 0) and f(0, 1)

= ax + by

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f(x, y) = ax + by

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“Rows reach right,

columns lean left.”

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cosine cosign

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Vectors are detectors and emitters.

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←

=

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→

=

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→

=

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←

=

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f(x, y) = ax + by,

f(x, y) = cx + dy

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row

vectors

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np.linalg.svd(A)

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x = np.linalg.solve(A, b)

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x = np.linalg.lstsq(A, b)[0]

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​

Apple

Bread

Cake

Alan

4

3

2

Betty

5

3

1

Carlos

1

3

5

Dana

2

3

4

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​

Apple

Bread

Cake

Alan

1

0

-1

Betty

2

0

-2

Carlos

-2

0

2

Dana

-1

0

1

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​

Apple

Bread

Cake

Apple

10

0

-10

Bread

0

0

0

Cake

-10

0

10

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​

Apple

Bread

Cake

Alan

1/√2

0

-1/√2

Betty

1/√2

0

-1/√2

Carlos

-1/√2

0

1/√2

Dana

-1/√2

0

1/√2

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​

Alan

Betty

Carlos

Dana

Alan

1

1

-1

-1

Betty

1

1

-1

-1

Carlos

-1

-1

1

1

Dana

-1

-1

1

1

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For square matrices

maybe:

np.linalg.eig(A)

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planspace.org

ajschumacher@gmail.com

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End