What is a matrix of format (n,p) over K?
A rectangular table with n rows and p columns whose entries belong to K.
What does K denote in a matrix over K?
The set of real numbers R or complex numbers C.
What defines a square matrix of order n?
A matrix with n rows and n columns.
What is a symmetric matrix?
A square matrix M where Mij equals Mji for every pair (i,j).
What is the identity matrix In of order n?
A diagonal matrix with all diagonal entries equal to 1.
When does the product AB of matrices A and B exist?
Only when A's columns equal B's rows.
What formula defines the entries of the matrix product AB?
What is the transpose of a matrix A in Mnp(K)?
A matrix in Mpn(K) whose rows are A's columns.
How is the trace of a square matrix A defined?
As the sum of its diagonal entries.
What is the formula for the trace of matrix A?
Which properties does matrix multiplication satisfy?
Associativity and distributivity.
What does the identity matrix satisfy in multiplication with A?
What is the formula for the determinant of a 2-by-2 matrix?
\det\begin{pmatrix}a&b\c&d\end{pmatrix}=ad-bc
How is a 3-by-3 determinant expanded?
By expressing it as signed entries multiplied by their corresponding minors along a row or column.
What is the minor of an entry in a square matrix?
The determinant of the submatrix after deleting row i and column j.
What equals the determinant of a triangular matrix?
The product of its diagonal entries.
What is the determinant of the identity matrix ?
What happens to a determinant when two rows or columns are exchanged?
It is multiplied by -1.
What effect does adding a linear combination of other rows or columns to one row or column have on the determinant?
It leaves the determinant unchanged.
What condition defines an invertible square matrix A?
There exists a square matrix B such that AB = In = BA.
What notation is used for the inverse matrix B of A?
It is written as .
When is a square matrix A invertible in terms of its determinant?
If and only if det(A) ≠ 0.
How is the cofactor αij of entry (i,j) defined?
αij times its minor Δij.
What is the formula for the inverse of an invertible square matrix A?
A^{-1} = \frac{1}{\det A} \; .
What defines a linear system in terms of equations and unknowns?
It is a system of n linear equations in p unknowns with coefficients and right-hand sides in K.
How is a linear system expressed in matrix form?
As AX = B, with A the coefficient matrix, X the unknown vector, and B the right-hand-side vector.
What is the rank of a matrix?
The maximum size of a square submatrix with nonzero determinant.
How is the rank of a linear system defined?
It is the rank of its coefficient matrix.
What happens if the system rank r is less than the number of unknowns p?
Then p−r unknowns are treated as parameters.
What happens if the system rank r is greater or equal to the number of unknowns p?
The unknowns are determined from p equations and checked against the remaining ones.
What is the main method of Gaussian elimination for solving linear systems?
Using elementary row operations to make the coefficient matrix triangular, then solving upward.
What formula gives the solution for each unknown in a Cramer system?
where replaces column i of A by B.
What is an internal composition law on a nonempty set E?
It is an application from E × E into E.
What is the pair (E,⋆) called when ⋆ is an internal composition law on E?
It is called a magma.
When is a subset A of E stable under an internal law ⋆?
When for every x,y in A, x⋆y belongs to A.
What condition defines associativity for a law ⋆ on E?
a⋆(b⋆c) equals (a⋆b)⋆c for every a,b,c in E.
What does an identity element e for ⋆ satisfy?
a⋆e = a = e⋆a for every a in E.
When is a law ⋆ on E commutative?
When x⋆y equals y⋆x for every x,y in E.
What defines a group in algebra?
A nonempty set with an associative law, identity, and inverses for all elements.
When is a group called abelian?
When its internal law is commutative.
What is a subgroup of a group G?
A subset of G with a group structure under G's law.
What conditions characterize a subgroup H of (G,⋆)?
H is nonempty, stable under ⋆, contains identity, and inverses of its elements.
What property defines a group morphism f from (G,•) to (F,⋆)?
It satisfies f(x•y) = f(x)⋆f(y) for all x,y in G.
What identities does a group morphism preserve?
It maps G's identity to F's identity and inverses to inverses.
What is the kernel of a group morphism f from G to F?
The set of elements in G mapped to F's identity.
What is the image of a group morphism f from G to F?
The set of values f(x) for all x in G.
What defines a subgroup H of a group G?
H is a nonempty subset with group structure under G's induced law.
What three conditions characterize a subgroup H of a group G?
H is closed under the group law, contains the identity, and contains inverses.
When is a subset H of G a subgroup using the condition on x and y in H?
If H is nonempty and for all x,y in H, x⋆y⁻¹ is in H.
What chains of subgroups are given by the inclusions involving (ℤ,+), (ℚ,+), and (ℝ,+)?
(ℤ,+)⊂(ℚ,+)⊂(ℝ,+) form a chain of subgroups.
What chains of subgroups are given by the inclusions involving (ℚ*,×), (ℝ*,×), and (ℂ*,×)?
(ℚ*,×)⊂(ℝ*,×)⊂(ℂ*,×) form a chain of subgroups.
What condition defines a group morphism f:G→F?
f(x•y) = f(x)⋆f(y) for all x,y in G.
What does a group morphism do to the identity element of G?
It sends the identity of G to the identity of F.
What does a group morphism do to the inverse of an element x in G?
It sends the inverse of x to the inverse of f(x).
What is true about the composition of two group morphisms?
Their composition is again a group morphism.
How is the kernel of a group morphism f:G→F defined?
It is the set of elements x in G with f(x) equal to the identity in F.
How is the image of a group morphism f:G→F defined?
It is the set of all f(x) for x in G, a subset of F.
When is a group morphism f:G→F injective?
If and only if its kernel is {eG}.
When is a group morphism f:G→F an isomorphism?
If and only if it is both injective and surjective.
What is a K-vector space in terms of addition and scalar multiplication?
A K-vector space is a nonempty set with addition forming an abelian group and scalar multiplication satisfying distributivity, associativity, and 1•u = u.
Which scalars are used in the course's vector spaces?
The scalars are either ℝ or ℂ.
What are the elements of a K-vector space called?
They are called vectors.
Name three examples of vector spaces given in the course.
ℝⁿ, Mₙₚ(ℝ), and ℂⁿ are examples of vector spaces.
Over which fields is ℂⁿ a vector space?
ℂⁿ is a vector space over both ℂ and ℝ.
How is addition defined in ℝⁿ?
Addition in ℝⁿ is defined componentwise by adding corresponding components.
How is scalar multiplication defined in ℝⁿ?
Scalar multiplication in ℝⁿ is defined componentwise by multiplying each component by the scalar.
What defines a vector u as a linear combination of vectors u₁,…,uₚ?
There exist scalars λ₁,…,λₚ such that u=λ₁u₁+⋯+λₚuₚ.
When is a family of vectors spanning for a vector space E?
Every vector of E is a linear combination of the vectors in the family.
What condition makes a family (u₁,…,u_q) linearly independent?
λ₁u₁+⋯+λ_qu_q=0 implies λ₁=⋯=λ_q=0.
When is a family of vectors linearly dependent?
If one vector is a linear combination of the others.
What two properties define a basis of a K-vector space E?
Being both linearly independent and spanning.
What uniqueness property holds for vector representations in a basis B=(e₁,…,eₙ)?
Every vector u∈E has a unique representation u=α₁e₁+⋯+αₙeₙ.
How is the dimension of a vector space defined?
It is the number of vectors in any one of its bases.
What defines a vector subspace in a K-vector space E?
It is a nonempty subset closed under vector addition and scalar multiplication.
What condition characterizes a vector subspace using linear combinations?
It is nonempty and closed under all linear combinations αu+βv with u,v in F and α,β in K.
What is the subspace generated by a set A={u₁,…,uₚ}?
The set of all linear combinations of the vectors in A, denoted vect A.
What formula relates the dimension of the sum of two finite-dimensional subspaces E₁ and E₂?
When are two subspaces E₁ and E₂ called supplementary?
When their sum is E and their intersection is the zero vector, written .
How is the rank of a family A defined?
It is the dimension of the subspace generated by A, rank A = dim(vect A).
Test your knowledge with 59 questions on Group and Vector Space Algebra.
1. A matrix of format over has which dimensions?
2. Which condition characterizes a square matrix of order ?
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