Linear Algebra Lecture Review Flashcards

Key definitions, theorems, and procedures from the Linear Algebra I lecture organised as Q&A flashcards for exam practice.

What is a vector in ℝⁿ and how are its components denoted?

An ordered n-tuple x = (x₁,…,xₙ)ᵀ with each xᵢ ∈ ℝ; x₁,…,xₙ are its components.

How are vector addition and scalar multiplication defined in ℝⁿ?

x + y = (x₁ + y₁,…,xₙ + yₙ)ᵀ and λx = (λx₁,…,λxₙ)ᵀ for x,y ∈ ℝⁿ, λ ∈ ℝ.

Name three fundamental algebraic laws satisfied by vector addition and scalar multiplication.

Associative and commutative laws of addition, and distributive laws: λ(x+y)=λx+λy, (λ+µ)x=λx+µx.

What is a linear combination of vectors v₁,…,vₘ?

Any sum λ₁v₁ + … + λₘvₘ with scalars λ₁,…,λₘ ∈ ℝ (or the underlying field).

Define Span(v₁,…,vₘ).

The set of all linear combinations of v₁,…,vₘ; it is the subspace they generate.

What is an m×n matrix?

A rectangular array A = (a{jk}) with m rows, n columns, entries a{jk} in ℝ (or K).

Describe a diagonal, an upper triangular, and a lower triangular matrix.

Diagonal: non-zero only on main diagonal. Upper triangular: non-zero on/above diagonal. Lower triangular: non-zero on/below diagonal.

How are matrix addition and scalar multiplication defined?

Entry-wise: (A+B){jk}=a{jk}+b{jk}; (λA){jk}=λa_{jk}.

Give the formula for the matrix-vector product A x where A ∈ ℝ^{m×n}.

(Ax)j = Σ{k=1}^n a{jk} xk for j = 1,…,m producing a vector in ℝᵐ.

Why is the map x ↦ Ax linear?

It satisfies A(x+y)=Ax+Ay and A(λx)=λAx for all vectors and scalars.

Interpret Ax as a combination of A’s columns.

Ax = x₁A₁ + … + xnAn, i.e., a linear combination of the column vectors of A.

Write a linear system in matrix form.

Ax = b with A the coefficient matrix, x the unknown vector, b the right-hand side.

What are the dimensions of the product B A when B ∈ ℝ^{k×m}, A ∈ ℝ^{m×n}?

k×n; entry c{ji}=Σ{ℓ=1}^m b{jℓ} a{ℓi}.

State the identity matrix and its key property.

Eₙ with 1’s on the diagonal, 0 elsewhere; for any A, AEₙ = A and EₘA = A.

Define the transpose of a matrix.

Aᵀ has entries (Aᵀ){jk} = a{kj}, swapping rows and columns.

When is a matrix symmetric?

If A = Aᵀ.

Give two basic identities for transposition.

(Aᵀ)ᵀ = A and (AB)ᵀ = BᵀAᵀ.

What is the canonical (Euclidean) inner product in ℝⁿ?

⟨x,y⟩ = Σ{k=1}^n xk y_k.

How is the Euclidean norm defined and list two properties.

‖x‖=√⟨x,x⟩; satisfies ‖λx‖=|λ|‖x‖ and the triangle inequality ‖x+y‖≤‖x‖+‖y‖.

State the Cauchy-Schwarz inequality.

|⟨x,y⟩| ≤ ‖x‖‖y‖ for all x,y.

When are two vectors orthogonal? What is the angle formula?

If ⟨x,y⟩=0; cos α = ⟨x,y⟩ / (‖x‖‖y‖).

Define a hyperplane in ℝⁿ.

H_{x,b} = { y ∈ ℝⁿ : ⟨x,y⟩ = b } with normal vector x ≠ 0.

What is a subspace of ℝⁿ?

A non-empty set U closed under vector addition and scalar multiplication.

What is the standard basis of ℝⁿ?

e₁=(1,0,…), e₂=(0,1,…), …, eₙ with a single 1 in each coordinate.

Define linear independence.

Vectors v₁,…,vm are independent if Σ λj vj = 0 implies all λj = 0.

What is a basis of a vector space V?

A set of linearly independent vectors whose span equals V.

Define the dimension of a vector space.

The number of vectors in any (every) basis; written dim V.

What is a linear map between vector spaces?

f satisfies f(v+w)=f(v)+f(w) and f(λv)=λf(v).

How is every linear map f:ℝⁿ→ℝᵐ represented?

By a unique matrix A with f(x)=Ax.

Define kernel and image of a matrix A.

ker A={x:Ax=0} ⊂ ℝⁿ; im A=Span(columns of A) ⊂ ℝᵐ.

What is the rank of a matrix and its link to the image?

Maximum number of independent columns; rank A = dim im A.

When is a square matrix invertible?

If there exists A⁻¹ with A⁻¹A = AA⁻¹ = E; equivalently, Ax=0 only for x=0.

State the LU (LR) decomposition idea.

For suitable A, find lower triangular L (diag 1) and upper triangular R with A=LR.

List the three elementary row operations used in Gaussian elimination.

Swap two rows; add λ times one row to another; multiply a row by non-zero λ.

How can row-echelon form reveal solvability of Ax=b?

If a zero row in A matches a non-zero entry in b, the system is inconsistent; otherwise solutions exist, unique when every column has a pivot.

What does Gram–Schmidt produce from any independent set?

An orthogonal (or orthonormal) basis spanning the same subspace.

Give the orthogonal projection of x onto a subspace V with ONB (v₁,…,v_k).

PV(x) = Σ{j=1}^k ⟨vj,x⟩ vj = U Uᵀ x where U=(v₁…v_k).

State the QR decomposition of a full-rank A ∈ ℝ^{m×n}.

A = Q R with Q having orthonormal columns and R upper triangular with positive diagonal.

Provide the normal-equation solution of min‖Ax−b‖₂.

x* = (AᵀA)⁻¹ Aᵀ b when A has full column rank.

Define eigenvalue and eigenvector of A.

λ ∈ ℂ is an eigenvalue if ∃ v≠0 with Av = λv; v is the corresponding eigenvector.

What is the characteristic polynomial of A?

P_A(λ)=det(A−λE); its zeros are the eigenvalues.

State the spectral theorem for real symmetric matrices.

A symmetric A ∈ ℝ^{n×n} has n real eigenvalues and an orthonormal basis of eigenvectors; A = Q D Qᵀ.

What are singular values and the SVD of A ∈ ℝ^{m×n}?

σ₁≥…≥σ_r>0 are √eigenvalues of AᵀA; A = U Σ Vᵀ with U,V orthonormal, Σ diagonal (possibly rectangular).

What is a vector in ℝ^n and how are its components denoted?

An ordered n-tuple x = (x1,…,xn)^T with each xi ∈ ℝ; x1,…,x_n are its components.

How are vector addition and scalar multiplication defined in ℝ^n?

x + y = (x1 + y1,…,xn + yn)^T and λx = (λx1,…,λxn)^T for x,y ∈ ℝ^n, λ ∈ ℝ.

Name three fundamental algebraic laws satisfied by vector addition and scalar multiplication.

Associative and commutative laws of addition, and distributive laws: λ(x+y)=λx+λy, (λ+µ)x=λx+µx.

What is a linear combination of vectors v1,…,vm?

Any sum λ1v1 + … + λmvm with scalars λ1,…,λm ∈ ℝ (or the underlying field).

Define Span(v1,…,vm).

The set of all linear combinations of v1,…,vm; it is the subspace they generate.

What is an m×n matrix?

A rectangular array A = (a{jk}) with m rows, n columns, entries a{jk} in ℝ (or K).

Describe a diagonal, an upper triangular, and a lower triangular matrix.

Diagonal: non-zero only on main diagonal. Upper triangular: non-zero on/above diagonal. Lower triangular: non-zero on/below diagonal.

How are matrix addition and scalar multiplication defined?

Entry-wise: (A+B){jk}=a{jk}+b{jk}; (λA){jk}=λa_{jk}.

Give the formula for the matrix-vector product A x where A ∈ ℝ^{m×n}.

(Ax)j = Σ{k=1}^n a{jk} xk for j = 1,…,m producing a vector in ℝ^m.

Why is the map x ↦ Ax linear?

It satisfies A(x+y)=Ax+Ay and A(λx)=λAx for all vectors and scalars.

Interpret Ax as a combination of A’s columns.

Ax = x1A1 + … + xnAn, i.e., a linear combination of the column vectors of A.

Write a linear system in matrix form.

Ax = b with A the coefficient matrix, x the unknown vector, b the right-hand side.

What are the dimensions of the product B A when B ∈ ℝ^{k×m}, A ∈ ℝ^{m×n}?

k×n; entry c{ji}=Σ{ℓ=1}^m b{jℓ} a{ℓi}.

State the identity matrix and its key property.

En with 1’s on the diagonal, 0 elsewhere; for any A, AEn = A and E_mA = A.

Define the transpose of a matrix.

A^T has entries (A^T){jk} = a{kj}, swapping rows and columns.

When is a matrix symmetric?

If A = A^T.

Give two basic identities for transposition.

(A^T)^T = A and (AB)^T = B^TA^T.

What is the canonical (Euclidean) inner product in ℝ^n?

⟨x,y⟩ = Σ{k=1}^n xk y_k.

How is the Euclidean norm defined and list two properties.

‖x‖=\sqrt{⟨x,x⟩}; satisfies ‖λx‖=|λ|‖x‖ and the triangle inequality ‖x+y‖≤‖x‖+‖y‖.

State the Cauchy-Schwarz inequality.

|⟨x,y⟩| ≤ ‖x‖‖y‖ for all x,y.

When are two vectors orthogonal? What is the angle formula?

If ⟨x,y⟩=0; \cos α = ⟨x,y⟩ / (‖x‖‖y‖).

Define a hyperplane in ℝ^n?

H_{x,b} = { y ∈ ℝ^n : ⟨x,y⟩ = b } with normal vector x ≠ 0.

What is a subspace of ℝ^n?

A non-empty set U closed under vector addition and scalar multiplication.

What is the standard basis of ℝ^n?

e1=(1,0,…), e2=(0,1,…), …, e_n with a single 1 in each coordinate.

Define linear independence.

Vectors v1,…,vm are independent if Σ λj vj = 0 implies all λ_j = 0.

What is a basis of a vector space V?

A set of linearly independent vectors whose span equals V.

Define the dimension of a vector space.

The number of vectors in any (every) basis; written dim V.

What is a linear map between vector spaces?

f satisfies f(v+w)=f(v)+f(w) and f(λv)=λf(v).

How is every linear map f:ℝ^n→ℝ^m represented?

By a unique matrix A with f(x)=Ax.

Define kernel and image of a matrix A.

ker A={x:Ax=0} ⊂ ℝ^n; im A=Span(columns of A) ⊂ ℝ^m.

What is the rank of a matrix and its link to the image?

Maximum number of independent columns; rank A = dim im A.

When is a square matrix invertible?

If there exists A⁻¹ with A⁻¹A = AA⁻¹ = E; equivalently, Ax=0 only for x=0.

State the LU (LR) decomposition idea.

For suitable A, find lower triangular L (diag 1) and upper triangular R with A=LR.

List the three elementary row operations used in Gaussian elimination.

Swap two rows; add λ times one row to another; multiply a row by non-zero λ.

How can row-echelon form reveal solvability of Ax=b?

If a zero row in A matches a non-zero entry in b, the system is inconsistent; otherwise solutions exist, unique when every column has a pivot.

What does Gram–Schmidt produce from any independent set?

An orthogonal (or orthonormal) basis spanning the same subspace.

Give the orthogonal projection of x onto a subspace V with ONB (v1,…,vk).

PV(x) = Σ{j=1}^k ⟨vj,x⟩ vj = U U^T x where U=(v1…vk).

State the QR decomposition of a full-rank A ∈ ℝ^{m×n}.

A = Q R with Q having orthonormal columns and R upper triangular with positive diagonal.

Provide the normal-equation solution of min‖Ax−b‖₂.

x^* = (A^TA)⁻¹ A^T b when A has full column rank.

Define eigenvalue and eigenvector of A.

λ ∈ ℂ is an eigenvalue if ∃ v≠0 with Av = λv; v is the corresponding eigenvector.

What is the characteristic polynomial of A?

P_A(λ)=det(A−λE); its zeros are the eigenvalues.

State the spectral theorem for real symmetric matrices.

A symmetric A ∈ ℝ^{n×n} has n real eigenvalues and an orthonormal basis of eigenvectors; A = Q D Q^T.

What are singular values and the SVD of A ∈ ℝ^{m×n}?

σ1≥…≥σr>0 are \sqrt{\text{eigenvalues of } A^TA}; A = U Σ V^T with U,V orthonormal, Σ diagonal (possibly rectangular).