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1 Chapter 1
1.1.7: Ifx,⃗y,⃗w⃗ ∈ R n^ and c, d ∈ R:
- V1:x⃗ +y⃗ ∈ R n
- V2: (x⃗ +y⃗ ) +w⃗ =x⃗ + (y⃗ +w⃗ )
- V3:y⃗ +x⃗ =x⃗ +y⃗
- V4: There exists a vector ⃗ 0 ∈ R n^ such that x⃗ + ⃗0 =x⃗ for allx⃗ ∈ R n
- V5: For every vectorx⃗ ∈ R n , there exists −x⃗ wherex⃗ + (−x⃗ ) = ⃗ 0
- V6: xc⃗ ∈ R n
- V7: c(d⃗x ) = (cdx)⃗
- V8: (c + dx)⃗ = xc⃗ + d⃗x
- V9: (x⃗ +y⃗ )c = xc⃗ + c⃗y
- V10: 1x⃗ =x⃗
1.2.13: A set of vectors {v ⃗ 1 , ...,⃗v (^) k } is linearly dependent if and only if v ⃗ i ∈ Span{v ⃗ 1 , ...⃗v (^) i − 1 ,⃗v (^) i +1, ...,⃗v (^) k } for some i, 1 ≤ i ≤ k. 1.2.20: if one set of vectors {v ⃗ 1 , ...,⃗v (^) k } contains the zero vector, then it is linearly dependent. 1.2.28: If β = {v ⃗ 1 , ...,⃗v (^) k } is a basis for a subset S of R n , then every vectorx⃗ ∈ S can be written as a unique linear combination of the vectors in β. 1.3.2 (Subspace Test): Let S be a non-empty subset of R n. Ifx⃗ +y⃗ ∈ S and c⃗x ∈ S for allx,⃗⃗y ∈ S and c ∈ R, then S is a subspace of R n 1.3.9: Ifv ⃗ 1 , ...,⃗v (^) k , then S = Span{v ⃗ 1 , ...,⃗v (^) k } is a subspace of R n 1.4.1: Ifx,⃗⃗y ∈ R^2 and θ is the angle betweenx⃗ andy⃗ thenx⃗ ·y⃗ = ∥x⃗ || ||y⃗ || cos x
1.4.4: Ifx,⃗⃗y,⃗z ∈ R n^ and s, t ∈ R, then: 1.x⃗ ·x⃗ ≥ 0 andx⃗ ·x⃗ = 0 if and only ifx⃗ = ⃗ 0 2.x⃗ ·y⃗ =y⃗ ·x⃗ 3.x⃗ · (s⃗y + zt⃗ ) = sx(⃗ ·y⃗ ) + tx(⃗ ·z⃗ ) 1.4.8: Ifx,⃗⃗y ∈ R n^ and c ∈ R, then:
- ||x|| ≥ 0 and ||x|| = 0 if and only if x⃗ = 0 2.||c⃗x || = |c| || x⃗||
- ||x⃗ ·y⃗ || ≤ ||x⃗ || ||y⃗ || (Cauchy-Schwartz Inequality)
- ||x⃗ +y⃗ || ≤ ||x⃗ || + ||y⃗ || (Triangle Inequal- ity) 1.4.17: Suppose thatv,⃗⃗w,⃗x ∈ R^3 and c ∈ R
- Ifn⃗ =v⃗ ×w⃗ then for anyy⃗ ∈ Span{v,⃗⃗w } we havey ⃗ ·n⃗ = 0 2)v⃗ ×w⃗ = −w⃗ ×v⃗ 3)v⃗ ×v⃗ = ⃗ 0 4)v⃗ ×w⃗ = ⃗0 if and only if eitherv⃗ = ⃗0 or w⃗ is a scaler multiple ofv⃗ 5)v⃗ × w(⃗ +x⃗ ) =v⃗ ×w⃗ +v⃗ ×x⃗
- (vc⃗ ) × (wc⃗ ) = cv(⃗ ×w⃗ )
- ||v⃗ ×w⃗ || = ||v⃗ || ||w⃗ || | sin θ| where θ is the angle betweenv⃗ andw⃗
1.4.18: Letv,⃗⃗w,⃗ b ∈ R^3 with {v,⃗⃗w } being linearly independent and let P be a plane in R^3 with vector equationx⃗ = s⃗v + wt⃗ +⃗ b, s.t ∈ R. Ifn⃗ =v⃗ ×w⃗ , then the equation for the plane is (x⃗ − ⃗b) ·n⃗ = 0
2 Chapter 2
Theorem 2.1.11: If the system of linear equations:
a 11 x 1 + ... + a 1 n x n = b 1 a 21 x 2 + ... + a 2 n x n = b 2
... ... a m 1 x 1 + ... + a mn x n = b m
has two distinct solutionss⃗ =
s 1 . . s n
and ⃗t =
t 1 . . t n
then for every c ∈ R,⃗ s + cs(⃗ − t⃗) is a solu- tion, and furthermore these solutions are all distinct. 2.2.11: If the augmented matrices
[
A 1 | b⃗ 1
]
and [ A 2 | b⃗ 2
]
are row equivalent, then the system of linear equations associated with each augmented ma- trix are equivalent. 2.2.13: If A is a matrix, then A has a unique reduced row echelon form of R. 2.2.27: The solution set of a homogeneous system of m linear equations and in n variables is a subspace of R n 2.2.30: Let S b be the solution set of the system [ A | ⃗b ] and let S 0 be the solution set of the as- sociated homogeneous system [ A | ⃗0 ]. Then ifx ⃗ p is any particular solution in S b.
S b = {x ⃗ p +s⃗ |s⃗ ∈ S 0 }
2.3.4: For any m × n matrix A we have
rank A ≤ min(m, n)
2.3.5 (System Rank Theorem): Let A be the coefficient matrix of a system of m linear equations and in n unknowns [ A | ⃗b ].
- The system [ A | ⃗b ] is inconsistent if and only if
rank A < [ A | ⃗b ]
- If the system [ A | ⃗b ]. is consistent, then the system contains (n − rank A) free variables 2.4.3: Let {v ⃗ 1 , ...,⃗v (^) k } be a set of vectors in R n^ and let A = v[⃗ 1 , ...,⃗v (^) k ]. Then {v ⃗ 1 , ...,⃗v (^) k } is linearly independent if and only if rank A = k. 2.4.8: Let {v ⃗ 1 , ...,⃗v (^) k } be a set of vectors in R n^ and let A = [v⃗ 1 , ...,⃗v (^) k ]. Then {v ⃗ 1 , ...,⃗v (^) k } spans R n^ if and only if rank A = n 2.4.11: A set of n vectors {v ⃗ 1 , ...,⃗v (^) k } in R n^ is linearly independent if and only if it spans R n
3 Chapter 3
3.1.4: If A, B, C ∈ M m × n (R) and c, d ∈ R then
- A + B ∈ M m × n (R)
- (A + B) + C = A + (B + C)
- A + B = B + A
- There exists a matrix O m,n ∈ M m × n (R), such that A + (O m × n ) = A for all A
- For every A ∈ M m × n (R) there exists (−A) ∈ M m × n (R) such that A + (−A) = O m,n
- cA ∈ M m × n (R)
- c(dA) = (cd)A
- (c + d)A = cA + dA
- c(A + B) = cA + cB
- 1 A = A
3.1.7: Let A, B ∈ M m × n (R) and c ∈ R, then
- (A T^ ) T^ = A
- (A + B) T^ = A T^ + B T
- (cA) T^ = cA T
3.1.19 (Column Extraction Theorem): Ife ⃗ i is the i-th standard basis vector for A = [a ⃗ 1 ...⃗a (^) n ], then A⃗e (^) i =a⃗ (^) i 3.1.20: Ifx,⃗⃗y ∈ R n , thenx⃗ T y⃗ =x⃗ ·y⃗ 3.1.22: If A ∈ M m × n ,⃗x,⃗y ∈ R n^ and c ∈ R, then
- Ax(⃗ +y⃗ ) = A⃗x + A⃗y
- A(c⃗x ) = c(A⃗x )
3.1.29: If A, B and C are matrices of the correct size so that the required product is defined and t ∈ R, then
- A(B + C) = AB + AC
- (A + B)C = AC + BC
- t(AB) = (tA)B = A(tB)
- A(BC) = (AB)C
- (AB) T^ = B T^ A T
3.1.32 (Matrix Equality Theorem): If A and B are m × n matrices such that A⃗x = B⃗x for every x ∈ R n , then A = B. 3.1.32: If I = [e⃗ 1 ...⃗e (^) n ], then for any n × n matrix A, we have
AI = A = IA
3.2.2 If A is an m × n matrix and f : R n^ → R m is defined by f x(⃗ ) = A⃗x , then for allx,⃗⃗y ∈ R n^ and b, c ∈ R we have
f (xb⃗ + yc⃗ ) = bf x(⃗ ) + cf y(⃗ )
3.2.8: If L : R n^ → R m^ is a linear mapping, then L(⃗ 0) = ⃗ 0 3.2.9: Every linear mapping L : R n^ → R m^ can be represented as a linear mapping with matrix whose i − th column is the image of the i − th standard basis vector of R n^ under L for all 1 ≤ i ≤ n. That is, Lx(⃗ ) = [Lx]⃗ where
[L] = [Le(⃗ 1 ) ... Le(⃗ (^) n )]
3.2.15: If R θ : R^2 → R^2 is a rotation with rotation A = [R θ ], then the columns of A are orthogonal unit vectors. 3.3.6: If L : R n^ → R m^ is a linear mapping, then Range(L) is a subspace of R m 3.3.14: If L : R n^ → R m^ is a linear mapping, then Ker(L) is a subspace of R n 3.3.16: Let L : R n^ → R m^ be a linear mapping. L is one-to-one if and only if for everyu,⃗⃗v ∈ R n^ such that Lu(⃗ ) = Lv(⃗ ), we must haveu⃗ =v⃗ 3.3.18: Let L : R n^ → R m^ be a linear mapping with standard matrix [L]. Then,x⃗ ∈ Ker(L) if and only if [Lx]⃗ = ⃗ 0 3.3.19: Let A ∈ M m × n (R). The set {x⃗ ∈ R n^ | A⃗x = 0 } is a subspace of R n 3.3.23: Let A be an m×n matrix. Suppose the vector equation of the solution set A⃗x = ⃗0 as determined by the Gauss-Jordan algorithm is given by
x⃗ = t 1 v⃗ 1 + ... + t k v⃗ (^) k , t 1 , ..t k ∈ R
Then, {v⃗ 1 , ...,⃗v (^) l } is a basis for N ull(A) 3.3.24: If A is an m × n matrix, then
dim N ull(A) = n − rank(A)
3.3.27: If L : R n^ → R m^ is a linear mapping with standard matrix [L] = A = [a⃗ 1 ...⃗a (^) n ], then
Range(L) = Span{a⃗ 1 , ...,⃗a (^) n }
4.2.5: If E is an elementary matrix, then E is invert- ible and E−^1 is also an elementary matrix. 4.2.8: If A is an m × n matrix and E is an m × m ele- mentary matrix corresponding to the row operation R i + cR j for i ̸= j then EA is the matrix obtained from A by performing the row operation R i + cR j on A 4.2.9: If A is an m × n matrix and E is an m × m elementary matrix corresponding to the row opera- tion cR i then EA is the matrix obtained from A by performing the row operation cR i on A 4.2.10: If A is an m×n matrix and E is an m×m el- ementary matrix corresponding to the row operation R i ↔ R j for i ̸= j then EA is the matrix obtained from A by performing the row operation R i ↔ R j on A 4.2.11: If A is an m × n matrix and E is an m × m elementary matrix, then
rank(EA) = rank(A)
4.2.12: If A is an m × n matrix with reduced eche- lon form R, then there exists a sequence E 1 , ..., E k of m × m elementary matrices such that E k ...E 2 E 1 A = R. In particular,
A = E− 1 1 E 2 − 1 ...E k −^1 R
4.2.17: If A is an n × n invertible matrix, then A and A−^1 can be written as a product of elementary matrices. 4.2.20: If E is an m × m elementary matrix, then E T^ is an elementary matrix.
4.3.11: Let A be an n × n matrix. For any i with 1 ≤ i ≤ n,
det A =
∑ n k =1 a ik C ik called the cofactor expansion across the i − th row. Or, for any j with 1 ≤ j ≤ n
det A =
∑ n k =1 a kj^ C kj is called the cofactor expansion across the j − th col- umn 4.3.16: If an n × n matrix A is upper or lower trian- gular, then
det A = a 11 a 22 ...a nn
4.3.18: If A is an n × n matrix and B is the ma- trix obtained from A by multiplying one row of A by c ∈ R, then det(B) = c det A 4.3.19: If A is an n × n matrix and B is the matrix obtained from A by swapping two rows of A, then det(B) = − det A 4.3.20: If an n × n matrix A has two identical rows, then det A = 0 4.3.21: If A is an n × n matrix and B is the matrix obtained from A by adding a multiple of one row to another row, then det B = det A 4.3.24: If A is an n × n matrix and E is an n × n elementary matrix, then det EA = det E det A 4.3.25 (Addition to the Invertible Matrix Theorem): An n × n matrix A is invertible if and only if det A ̸= 0 4.3.26: If A and B are n×n matrices, then det AB = det A det B 4.3.27: If A is an invertible matrix, then det A−^1 = 1 det A
4.3.28: If A is an n × n matrix, then det A = det A T 4.4.1: If A is an n × n matrix with cofactors C ij and i ̸= j, then ∑ n k =1(A) ik C jk^ = 0 4.4.2: If A is an invertible n × n matrix, then (A−^1 ) ij = (^) det^1 A C ji
5 Chapter 5
5.1.3: Every subspace S of R n^ has a basis 5.1.7: Let S is a subspace of R n. Let B = {v⃗ 1 , ...,⃗v (^) k } be a basis for S and let C = {w⃗ 1 , ...⃗w (^) n } be a set in S. If m > k, then C is linearly dependent. 5.1.8: If B = {v⃗ 1 , ...,⃗v (^) n } and C = {w⃗ 1 , ...,⃗w (^) n } are bases for S, then k = m 5.1.12: If dim S = k, then:
- A set of more than k vectors in S must be lin- early dependent.
- A set of vectors less that k vectors in S cannot span S
- A set of k vectors in S is linearly dependent if and only if it spans S
5.1.15: If S is an k-dimensional subspace of R n^ and {v⃗ 1 , ..,⃗v (^) l } is a linearly independent set in S with l < k, then there exist vectorsw⃗ (^) l +1, ...,⃗w (^) k in S such that {v⃗ 1 , ...,⃗v (^) l w,⃗ (^) l +1, ...,⃗w (^) n } is a basis for S. 5.1.16: Let S 1 and S 2 be subspaces of R n^ such that S 1 ⊆ S 2. Then dim S 1 ≤ dim S 2. Moreover, dim S 1 = dim S 2 if and only if dim S 1 = dim S 2
5.2.1: If B = {v⃗ 1 , ..,⃗v (^) k } is a basis for the subspace S of R n , then everyv⃗ ∈ S can be written as a unique linear combination of the vectors in B. 5.2.10: If S is a subspace of R n^ with basis B = {v⃗ 1 , ...,⃗v (^) k }, then for anyv,⃗⃗w ∈ S and s, t ∈ R we have
[s⃗v + wt⃗ ] B = sv[⃗ ] B + tw[⃗ ] B
5.2.16: If B and C are basis for k-dimensional sub- spaces S, then the change of coordinates matrices C P B and^ B P C satisfy
C P B B P C =^ I^ = B P C C P B
6 Chapter 6
6.1.7: If A and B are n × n matrices such that P −^1 AP = B for some invertible matrix P , then
- rank A = rank B
- det A = det B
- tr A = tr B where tr A =
∑ n i =1 a ii^ is called the trace of the matrix.
6.2.8: A scaler λ is an eigenvalue of an n × n matrix A if and only if C A (λ) = 0 6.2.13: If A is an n × n upper or lower triangular matrix, then the eigenvalues of A are the diagonal entries of A. 6.2.20: Let A and B are similar matrices, then A and B have the same characteristic polynomial, and hence the same eigenvalues. 6.2.21: If A is an n × n matrix with eigenvalue λ 1 , then 1 ≤ g λ 1 ≤ a λ 1 6.3.2 (Diagonalization Theorem): An n × n matrix A is diagonalizable if and only if there exists a basis {v⃗ 1 , ...,⃗v (^) n } for R n^ of eigenvalues of A.
6.3.3: If A is an n × n matrix with eigenpairs (λ 1 ,⃗v 1 ), (λ 2 ,⃗v 2 ), ..., (λ k v,⃗ (^) k ) where λ i ̸= λ j for i ̸= j, then {v⃗ 1 , ..,⃗v (^) l } is linearly independent. 6.3.4: If A is an n × n matrix with distinct eigenval- ues λ 1 , .., λ k and B i = {v⃗ (^) i, 1 , ...,⃗v (^) i,gλi } is a basis for the eigenspace λ i for 1 ≤ i ≤ k, then B 1 ∪B 2 ∪...∪B k is a linearly independent set. 6.3.5 (Diagonalizability Test): If A is an n×n matrix whose characterisitc polynomial factors as
C A (λ) = (λ − λ 1 ) aλ^1 ...(λ − λ k ) aλk
where λ 1 , ..., λ k are distinct eigenvalues of A, then A is diagonalizable if and only if g λi = a λi for 1 ≤ i ≤ k. 6.3.6: If A is an n × n matrix with n distinct eigen- values, then A is diagonalizable. 6.3.13: If λ 1 , .., λ n are all the n eigenvalues of an n × n matrix A, then
det A = λ 1 ...λ n and tr(A) = λ 1 + ... + λ n
6.4.1: Let A be an n × n matrix. If there ex- ists a matrix P and a diagonal matrix D such that P −^1 AP = D, then
A k^ = P D k P −^1
