Chapter 2 - Matrices I Flashcards
Explain what an m x n matrix is.
A rectangular array of m rows and n columns of numbers, known as entries.
Two matrices are equal if and only if [_____].
Two matrices are equal if and only if they are the same size and their corresponding entries are equal.
If A and B are matrices of the same size, the sum A + B is calculated by [_____].
If A and B are matrices of the same size, the sum A + B is calculated by adding corresponding entries.
If A is any matrix and k is any number, the scalar multiple kA is the matrix obtained from A by [_____].
If A is any matrix and k is any number, the scalar multiple kA is the matrix obtained from A by multiplying each entry of A by k.
If A, B and C are the same size,
A + B = [____] (Commutativity)
A + (B + C) = [____] (Associativity)
If A, B and C are the same size,
A + B = B + A (Commutativity)
A + (B + C) = (A + B) + C (Associativity)
Define the transpose of a matrix.
The transpose of a matrix is the matrix formed by swapping the rows and columns of the original matrix.
For any matrix A,
(A^T)^T = [\_\_\_] (kA)^T = [\_\_\_] (A+B)^T = [\_\_\_]
If A is an m x n matrix, A^T is an [____] matrix
For any matrix A,
(A^T)^T = A (kA)^T = kA^T (A+B)^T = A^T + B^T
If A is an m x n matrix, A^T is an n x m matrix.
What does it mean for a matrix to be symmetric?
A^T = A if and only if the matrix is symmetrical.
What is an ordered n-tuple?
An ordered sequence (a1, a2, a3, …, an) of real numbers.
What is meant by ℝ^n?
A set of all the ordered n-tuples from ℝ.
Given a system of linear solutions, we can consider the left hand side as [_____].
Given a system of linear solutions, we can consider the left hand side as a coefficient matrix A and the column x of variables, multiplied together: Ax.
Let A = [a1, a2, a3, …, an] be an n x m matrix, written in terms of its columns a1, a2, …, an.
If x = vector{x1, x2, …., xn} is any n-vector, the product Ax is defined to be the m-vector given by:
[___________________]
Let A = [a1, a2, a3, …, an] be an n x m matrix, written in terms of its columns a1, a2, …, an.
If x = vector{x1, x2, …., xn} is any n-vector, the product Ax is defined to be the m-vector given by:
Ax = x1a1 + x2a2 + x3a3 + … + xnan
Let A and B be m × n matrices, and let x and y be n-vectors in ℝ^n. Then,
A(x+y) = [\_\_\_\_\_] A(ax) = [\_\_\_\_\_] (A+B)x = [\_\_\_\_\_]
Let A and B be m × n matrices, and let x and y be n-vectors in ℝ^n. Then,
A(x+y) = Ax + Ay A(ax) = (aA)x (A+B)x = Ax + Bx
Let x1 be any particular solution to the system to Ax = b.
Then, every solution x2 to Ax = b has the form:
[_____] for some solution x0 of the associated homogeneous system Ax = 0.
Let x1 be any particular solution to the system to Ax = b.
Then, every solution x2 to Ax = b has the form:
x2 = x1 + x0 for some solution x0 of the associated homogeneous system Ax = 0.
Let Ax = b be a system of equations with augmented matrix [A | b]. Let rank A = r.
- rank [A | b] is [____]
- The system is consistent if and only if rank [A | b] = [____]
- The system is inconsistent if and only if rank [A | b] = [____]
Let Ax = b be a system of equations with augmented matrix [A | b]. Let rank A = r.
- rank [A | b] is r or r + 1
- The system is consistent if and only if rank [A | b] = r
- The system is inconsistent if and only if rank [A | b] = r + 1
Define the dot product of the ordered n-tuples, (a1, a2, … , an) and (b1, b2, … , bn).
a1b1 + a2b2 + … + anbn
Let A be an m × n matrix and let x be an n-vector. Then, each entry of the vector Ax is [_________].
Let A be an m × n matrix and let x be an n-vector. Then, each entry of the vector Ax is the dot product of the corresponding row of A with x.
Define the identity matrix, I.
The n x n matrix with 1s in the main diagonal, and zeros elsewhere.
Explain what is meant by the matrix transformation induced by A.
The transformation of a vector x, from ℝ^m to ℝ^n.
Let A and B be matrices of sizes m x n and n x k respectively. Then, the (i, j)-entry of AB is [_____].
Let A and B be matrices of sizes m x n and n x k respectively. Then, the (i, j)-entry of AB is the dot product of row i of A with column j of B.
Is matrix multiplication commutative?
No.
If a is any scalar, and AB and C are suitably sized matrices,
IA = AI = [\_\_\_\_], where I is the identity matrix. A(BC) = [\_\_\_\_] A(B+C) = [\_\_\_\_] (B+C)A = [\_\_\_\_] a(AB) = [\_\_\_\_] (AB)^T = [\_\_\_\_]
If a is any scalar, and AB and C are suitably sized matrices,
IA = AI = A, where I is the identity matrix. A(BC) = (AB)C A(B+C) = AB + AC (B+C)A = BA + CA a(AB) = (aA)B = A(aB) (AB)^T = B^T x A^T
Summarise the process of block partitioning a matrix in order to simplify matrix multiplication.
The matrix is divided up into ‘blocks’, that can be substituted for letters representing matrices. The matrix can be multiplied and resubstituted. The blocks must be compatible for multiplication.
Explain what a directed graph is.
A directed graph is a set of points (vertices) connected by arrows (edges). If the graph has n vertices [v1, v2, …, vn], then the adjacency matrix A = [a(ij)] is the n x n matrix whose (i,j)-entry is 1 if there is an edge from vj to vi and zero otherwise.
If A is the adjacency matrix of a directed graph with n vertices, then the (i, j)-entry of A^r is [_____]
If A is the adjacency matrix of a directed graph with n vertices, then the (i, j)-entry of A^r is the number of r-paths vj to vi.
