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Linear Algebra Cheatsheet: Basics of Vectors and Matrices, Summaries of Linear Algebra

Amridge UniversityLinear Algebra

Cheatsheet on Linear Algebra: Basics of vectors and matrices, Addition and scalar multiplication, Vector to matrix multiplication

Typology: Summaries

2019/2020

Uploaded on 10/09/2020

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Linear Algebra Cheatsheet
UiO Language Technology Group
1 Basics of vectors and matrices
1.1 Matrices
•Matrix is a rectangular 2-dimensional array of numbers (scalars).
•M∈Rm×nis a matrix Mwith mrows and ncolumns.
•For example:
M=
1234
0000
4321
•Here, Mis a 3×4matrix: it has 3 rows and 4 columns. 3and 4are the
dimensions of M.
1.2 Entries
•Matrices consist of entries.
•Mi,j or M[i,j]is the entry in the ith row and jth column of M.
•For example:
M0,0= 1
•NB: we use 0-indexed notation, following Python conventions.
1.3 Vectors
•Vector is a 1×nmatrix (NB: we use row vectors).
•v∈Rnis a vector vwith nentries or components (n-dimensional vector).
•For example:
v= [4,3,2,1]
•Here, vis a 4-dimensional vector.
•vior v[i]is the ith entry of the vector.
•For example:
v1= 3
1
pf3
pf4
pf5

Partial preview of the text

Download Linear Algebra Cheatsheet: Basics of Vectors and Matrices and more Summaries Linear Algebra in PDF only on Docsity!

Linear Algebra Cheatsheet

UiO Language Technology Group

1 Basics of vectors and matrices

1.1 Matrices

  • Matrix is a rectangular 2-dimensional array of numbers (scalars).
  • M ∈ Rm×n^ is a matrix M with m rows and n columns.
  • For example:

M =

  • Here, M is a 3 × 4 matrix: it has 3 rows and 4 columns. 3 and 4 are the dimensions of M.

1.2 Entries

  • Matrices consist of entries.
  • Mi,j or M[i,j] is the entry in the ith^ row and jth^ column of M.
  • For example: M 0 , 0 = 1
  • NB: we use 0-indexed notation, following Python conventions.

1.3 Vectors

  • Vector is a 1 × n matrix (NB: we use row vectors).
  • v ∈ Rn^ is a vector v with n entries or components (n-dimensional vector).
  • For example: v = [4, 3 , 2 , 1]
  • Here, v is a 4-dimensional vector.
  • vi or v[i] is the ith^ entry of the vector.
  • For example: v 1 = 3

2 Addition and scalar multiplication

2.1 Matrix addition

  • Matrix addition is simply adding the entries of two or more matrices one by one.
  • This summation results in another matrix:
  • M + M = M
  • For example:  
  • NB: we can add only matrices of the same dimensionality!
  • The resulting matrix retains the same dimensions ( 3 × 3 in the example above).
  • One can subtract matrices in the same way.

2.2 Multiplication by scalar

  • To multiply a matrix by scalar (a raw number), one also simply mul- tiplies all its entries by this scalar.
  • It results in another matrix of the same dimensionality.
  • For example: (^) 

 × 2 =

  • Note that the multiplication of a matrix by a scalar and the multiplication of a scalar by a matrix are equal:  

 × 2 = 2 ×

  • One can divide a matrix by a scalar in the same way:  
  • This essentially amounts to the scalar multiplication by fraction:  

 × 1

4.2 Process

  • The result of right-multiplying a vector v ∈ Rm^ (or, equally, v ∈ R^1 ×m) by a matrix W ∈ Rm×n^ is a vector y ∈ Rn - v · W = y - note how the matching dimensions m are ‘self-destroyed’.
  • Each component i of y is a sum of one-by-one multiplying columns of v by the entries of the ith^ column of W.
  • For example:

y = v · W = [2, 3] ·

[

]

= [17, 10 , 9]

  1. y 0 = v 0 × W 0 , 0 + v 1 × W 1 , 0 = 2 × 4 + 3 × 3 = 8 + 9 = 17
  2. y 1 = v 0 × W 0 , 1 + v 1 × W 1 , 1 = 2 × 2 + 3 × 2 = 4 + 6 = 10
  3. y 2 = v 0 × W 0 , 2 + v 1 × W 1 , 2 = 2 × 3 + 3 × 1 = 6 + 3 = 9
  • Here, the result is the 3-dimensional row vector y ∈ R^3.

5 Matrix to matrix multiplication

5.1 Matrix to matrix is another matrix

  • Any row vector is in fact a 1 × n matrix.
  • Thus, to multiply one matrix by another, is conceptually the same as multiplying a vector by a matrix.
  • Again, the number of columns in the left matrix W 1 must match the number of rows of the right matrix W 2 : W 1 ∈ Rm×n, W 2 ∈ Rn×z
  • But the result of this multiplication is another matrix : W 1 · W 2 = W 3 ∈ Rm×z - Again, the matching dimensions n are ‘self-destroyed’.

5.2 Process

  • For example, suppose W 1 ∈ R^2 ×^3 , W 2 ∈ R^3 ×^4 :

W 3 = W 1 · W 2 =

[

]

[

]

  • Here, W 3 ∈ R^2 ×^4
  • It is produced like this:
  • Each row i of W ^ is a product of multiplying the ith^ row of W ^ (a vector) by W ^ (a matrix):

1. W [0,:] = W [0,:] ·W ^ = [4, 2 , 3]·

 = [19, 17 , 19 , 16]

2. W [1,:] = W [1,:] ·W ^ = [3, 2 , 1]·

 = [11, 10 , 11 , 10]

  1. etc...

5.3 Properties of matrix multiplication

  • Matrix multiplication is not commutative: W 1 · W 2 6 = W 2 · W 1
  • Matrix multiplication is associative: W 1 · W 2 · W 3 = W 1 · (W 2 · W 3 ) = (W 1 · W 2 ) · W 3