• Substitution
① 4𝛼 + 1𝛽 + 1𝛾 = 9
o { ②1𝛼 + 2𝛽 + 3𝛾 = 14
③ 2𝛼 + 11𝛽 − 1𝛾 = 21
① 4𝛼 + 1𝛽 + 1𝛾 = 9
o → { 4② 4𝛼 + 8𝛽 + 12𝛾 = 56
2③ 4𝛼 + 22𝛽 − 2𝛾 = 42
① 4𝛼 + 1𝛽 + 1𝛾 = 9
o → {② − ① 0𝛼 + 7𝛽 + 11𝛾 = 47
③ − ① 0𝛼 + 21𝛽 − 3𝛾 = 33
① 4𝛼 + 1𝛽 + 1𝛾 = 9
o →{ ② 0𝛼 + 7𝛽 + 11𝛾 = 47
③ − 3② 0𝛼 + 0𝛽 − 36𝛾 = −108
−36𝛾 = −108 → 𝛾 = 3
o 7𝛽 + 33 = 47 → 𝛽 = 2
4𝛼 + 2 + 3 = 9 → 𝛼 = 1
• Points and Vectors
o Vector is coordinate – starting point
▪ 𝐴(𝑥1 , 𝑦1 )
▪ Through the origin gives
𝑥1
• 𝑎 = (𝑦 )
2
▪ Through 𝐵(𝑥2 , 𝑦2 ) gives
𝑥1 − 𝑥2
• 𝑎 = (𝑦 − 𝑦 )
1 2
• Subspaces
o Vector through the origin
o Properties subspace 𝑊
▪ Zero vector in 𝑊
▪ If 𝑥 and 𝑦 in 𝑊, then 𝑥 + 𝑦 in 𝑊
▪ If 𝑥 in 𝑊, then 𝜆𝑥 in 𝑊, with 𝜆 any real number
o Vectors must be independent
o Linear combinations
▪ 𝑥 = 𝛼𝑎 + 𝛽𝑏 + 𝛾𝑐 + ⋯ + 𝜔𝑧
o Vector independence
▪ 𝛼𝑎 + 𝛽𝑏 + 𝛾𝑐 = 0
• Independent if
𝛼=0
o {𝛽 = 0
𝛾=0
o Dimension is equal to the number of vectors in a basis
▪ All independent vectors in a plane
, • Matrix
o Matrices are depicted with an uppercase and vectors with a lowercase letter
o Multiple rows together
o Can be multiplied with a vector, if the number of rows of the vector is equal to the
number of columns of the matrix
o Transposed matrix
▪ First row becomes first column and vice versa
𝑎 𝑐 𝑎 𝑏
▪ 𝑀=( ) → 𝑀𝑇 = ( )
𝑏 𝑑 𝑐 𝑑
o Special Matrices
▪ Zero matrix
0 0
• 𝑂=( )
0 0
▪ Identity matrix
1 0 ⋯ 0
0 1 ⋯ 0
• 𝐼=( )
⋮ ⋮ ⋱ 0
0 0 0 1
• Multiplication with a vector gives the same vector as a result
o Algebraic rules
▪ Linearity
• 𝐴(𝑣 + 𝑤) = 𝐴𝑣 + 𝐴𝑤
• (𝐴 + 𝐵)𝑣 = 𝐴𝑣 + 𝐵𝑣
• 𝐴(𝜆𝑣) = (𝜆𝐴) = 𝜆(𝐴𝑣)
▪ For transposed matrices
• (𝐴𝑇 )𝑇 = 𝐴
• (𝐴 + 𝐵)𝑇 = 𝐴𝑇 + 𝐵𝑇
• (𝜆𝐴)𝑇 = 𝜆𝐴𝑇
o Algebraic rules for matrix products
▪ Rules for products
• (𝐷𝐴)𝑣 = 𝐷(𝐴𝑣)
• (𝐵𝐷)𝐴 = 𝐵(𝐷𝐴)
▪ Sum and product as with numbers
• 𝐴(𝐵 + 𝐶) = 𝐴𝐵 + 𝐴𝐶
▪ Deviant Rules
• Order matters
o 𝐴𝐵 ≠ 𝐵𝐴 in general
• For Transposed:
o (𝐴𝐵)𝑇 = 𝐵𝑇 ∙ 𝐴𝑇
▪ Reversed order
• Three forms of systems of equations
0.80𝑥 + 0.10𝑦 = 30
o
0.10𝑥 + 0.70𝑦 = 75
0.80 0.10 30
o ( )𝑥 + ( )𝑦 = ( )
0.10 0.70 45
0.80 0.10 𝑥 30
o ( ) ∙ (𝑦) = ( )
0.10 0.70 45
① 4𝛼 + 1𝛽 + 1𝛾 = 9
o { ②1𝛼 + 2𝛽 + 3𝛾 = 14
③ 2𝛼 + 11𝛽 − 1𝛾 = 21
① 4𝛼 + 1𝛽 + 1𝛾 = 9
o → { 4② 4𝛼 + 8𝛽 + 12𝛾 = 56
2③ 4𝛼 + 22𝛽 − 2𝛾 = 42
① 4𝛼 + 1𝛽 + 1𝛾 = 9
o → {② − ① 0𝛼 + 7𝛽 + 11𝛾 = 47
③ − ① 0𝛼 + 21𝛽 − 3𝛾 = 33
① 4𝛼 + 1𝛽 + 1𝛾 = 9
o →{ ② 0𝛼 + 7𝛽 + 11𝛾 = 47
③ − 3② 0𝛼 + 0𝛽 − 36𝛾 = −108
−36𝛾 = −108 → 𝛾 = 3
o 7𝛽 + 33 = 47 → 𝛽 = 2
4𝛼 + 2 + 3 = 9 → 𝛼 = 1
• Points and Vectors
o Vector is coordinate – starting point
▪ 𝐴(𝑥1 , 𝑦1 )
▪ Through the origin gives
𝑥1
• 𝑎 = (𝑦 )
2
▪ Through 𝐵(𝑥2 , 𝑦2 ) gives
𝑥1 − 𝑥2
• 𝑎 = (𝑦 − 𝑦 )
1 2
• Subspaces
o Vector through the origin
o Properties subspace 𝑊
▪ Zero vector in 𝑊
▪ If 𝑥 and 𝑦 in 𝑊, then 𝑥 + 𝑦 in 𝑊
▪ If 𝑥 in 𝑊, then 𝜆𝑥 in 𝑊, with 𝜆 any real number
o Vectors must be independent
o Linear combinations
▪ 𝑥 = 𝛼𝑎 + 𝛽𝑏 + 𝛾𝑐 + ⋯ + 𝜔𝑧
o Vector independence
▪ 𝛼𝑎 + 𝛽𝑏 + 𝛾𝑐 = 0
• Independent if
𝛼=0
o {𝛽 = 0
𝛾=0
o Dimension is equal to the number of vectors in a basis
▪ All independent vectors in a plane
, • Matrix
o Matrices are depicted with an uppercase and vectors with a lowercase letter
o Multiple rows together
o Can be multiplied with a vector, if the number of rows of the vector is equal to the
number of columns of the matrix
o Transposed matrix
▪ First row becomes first column and vice versa
𝑎 𝑐 𝑎 𝑏
▪ 𝑀=( ) → 𝑀𝑇 = ( )
𝑏 𝑑 𝑐 𝑑
o Special Matrices
▪ Zero matrix
0 0
• 𝑂=( )
0 0
▪ Identity matrix
1 0 ⋯ 0
0 1 ⋯ 0
• 𝐼=( )
⋮ ⋮ ⋱ 0
0 0 0 1
• Multiplication with a vector gives the same vector as a result
o Algebraic rules
▪ Linearity
• 𝐴(𝑣 + 𝑤) = 𝐴𝑣 + 𝐴𝑤
• (𝐴 + 𝐵)𝑣 = 𝐴𝑣 + 𝐵𝑣
• 𝐴(𝜆𝑣) = (𝜆𝐴) = 𝜆(𝐴𝑣)
▪ For transposed matrices
• (𝐴𝑇 )𝑇 = 𝐴
• (𝐴 + 𝐵)𝑇 = 𝐴𝑇 + 𝐵𝑇
• (𝜆𝐴)𝑇 = 𝜆𝐴𝑇
o Algebraic rules for matrix products
▪ Rules for products
• (𝐷𝐴)𝑣 = 𝐷(𝐴𝑣)
• (𝐵𝐷)𝐴 = 𝐵(𝐷𝐴)
▪ Sum and product as with numbers
• 𝐴(𝐵 + 𝐶) = 𝐴𝐵 + 𝐴𝐶
▪ Deviant Rules
• Order matters
o 𝐴𝐵 ≠ 𝐵𝐴 in general
• For Transposed:
o (𝐴𝐵)𝑇 = 𝐵𝑇 ∙ 𝐴𝑇
▪ Reversed order
• Three forms of systems of equations
0.80𝑥 + 0.10𝑦 = 30
o
0.10𝑥 + 0.70𝑦 = 75
0.80 0.10 30
o ( )𝑥 + ( )𝑦 = ( )
0.10 0.70 45
0.80 0.10 𝑥 30
o ( ) ∙ (𝑦) = ( )
0.10 0.70 45
