Terminology Flashcards
What are elementary matrices? How many types are there? Provide examples
Elementary matrices are obtained by performing a single elemntary row operation on an identity matrix, there are 3 types, row swap, row multiplication and row addition
How can you show two matrices A and B are row equivalent using elementary matrices?
Two matrices are equivalent if one can be transformed into the other using a sequence of elementary row operations
What is a non-singular matrix? Given a square matrix A, how can you determine whether A is non-singular or not?
A non singular matrix is a square matrix with a non zero determinant. To determine if singular compute det(A), if A doesn’t equal 0, A is non singular
Given a matrix An×m. What is the dimension of the transpose of A or A^T
? Explain why the two products AA^T and A^TA can be done.
If A is an nxm matrix, then it’s transpose A^t is an mxn matrix. AA^T & A^TA can be done because
A is nxm, and A^T is mxn
AA^T: (nxm)(mxn) is a valid product, result is nxn
A^TA: (mxn)(nxm) is a valid product result is mxm
What are Euclidean vector spaces? Provide a couple of examples of Euclidean vector
spaces.
Euclidian vector spaces are finite dimensional vector space with a notion of distance & angles defined with dot product.
ex. R^2 is a 2D real vector space
R^3 is a 3D real vector space
How can you show if a subset S is a subspace of a vector space V ?
A subset S is a subspace if the following properties hold:
Contains 0 vector
Closed under addition
Closed under scalar multiplication
Show that the set of vectors, S = {(x1, x2, x3)^T such that x1 = x2 and x3 = 0} is a subspace of R^3
Check subspace properties
0 vector
closed under addition
closed under scalar multiplication
What is the null space of a matrix, A? How can you find the null space of a matrix A.
The null space of a matrix is the set of all vectors x such that Ax=0.
To find null space:
Setup equation Ax=0
Form the augmented matrix
Reduce to RREF
Solve for free variable
Set of solutions form the null space
What is the span of a set of vectors, {v1, v2, . . . , vn}?
The span of vectors is the set of all linear combinations of these vectors
ex. span{v1, v2, …, vn} = {c1+v1, c2+v2, … + cn+vn | Ci element of R}
What is a spanning set for a Euclidean vector space, R^3?
A spanning set is a set of vectors whose span equals the whole space R^3
ex. {(1,0,0)^T, (0,1,0)^T, (0,0,1)^T} for R^3
Given 3 vectors {v1, v2, v3} ∈ R^3, provide 2 different methods which you can use to determine whether those 3 vectors are linearly independent or not.
Row reduction: After RREF if there is a pivot in each column it is linearly independent
Determinant: If matrix is square, if det doesn’t equal 0, it is linearly independent
What are the fundamental differences between a spanning set of vectors for R^3,
{v1, v2, v3}, and the span of a set of vectors, span{v1, v2, v3.}
A spanning set of vectors is a collection of vectors, the span is all possible linear combinations of those vectors
What is a basis of a vector space? What is the elementary basis of R^3? Give an example of a basis of R^3 that is not the elementary basis and explain why that set of vectors form a basis.
A basis of vector space is a set of linearly independent vectors that span the entire vector space.
An elementary basis of R^3 is {(1,0,0)^T, (0,1,0)^T,(0,1,1)^T}
This works because vectors span R^3 and are linearly independent
Given 3 vectors {v1, v2, v3} ∈ R^3, provide 2 different methods which you can use to
determine whether those 3 vectors form a basis of R^3.
Row Reduction: Row reduce to RREF, if 3 pivot columns are present it forms a basis
Determinant: (For 3x3) compute the det(A), if det(A) doesn’t equal 0 it forms a basis
What is the dimension of a vector space? What is the dimension of R^3? What is thedimension of a plane in R^3? Explain your reasoning.
The dimension of a vector space is the numnber of basis vectors required to span the space. The dimension of R^3 is 3 as it takes 3 independent vectors to span it. The dimension of a plane in R^3 is 2, as a plane can only be spanned by 2 linearly independent vectors
What is the rank of a matrix? Given a matrix A3×4, how can you find the rank of matrix A?
The rank of amtrix is the # of linearly independent rows or columns, it tell us the dimension of the column space. To find the rank of A3x4, Convert it to RREF and count the # of non zero rows, that count will be the rank
What is the row space and column space of a matrix A? How to find them? Are row space of A and RREF(A) the same? Are column space of A and RREF(A) the same? Why or why not?
Row space is the span of the row vectors of A. Colun space is the span of the coulumn vectors of A. To find row space, reduce to RREEF, the nonzero rows form a basis for the row space. For finding column space the pivot columns of A before reduction form a basis
Explain, step by step, how to demonstrate that a map L, which transforms vectors from vector space V to vector space W (denoted as L : V → W), is a linear transformation.
To show tha a map is a linear transformation verify that it satisfies 2 properties, additivity, and homogenity
Additivity: L(u+v) = L(u)+L(v)
Homogenity: L(cv) = cL(v)
Given a linear transformation L : R^n → R^m, what is the size of the matrix representationof L, A? How do you find A?
The matrix representation of L, A is of size: mxn because L maps R^n to R^m. To find A:
Apply L to the standard basis vectors of R^n, e1, e2, …, en.
The images L(e1), …, L(en) become the columns of A.
Given a linear transformation L : V → W, how do you find its Kernel and Range. Show that Kernel is a subspace of vector space V .
Kernel space is given by Ker(L) = {v elemt of V | L(V) = 0}
The range of L is Range(L) = {L(v) | v elemtn of V} W.
To find kernel solve L(v) = 0.
Range is getting all possible outputs L(V)
Kernel is a subspace because it contains the zero vector, is closed under addition, and is closed uinder scalar multiplication.
