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1 Prior work on matrix multiplication 2 Matrix multiplication is
... Editor: Kathy Cooper Given two n × n matrices A, B over {0, 1}, we define Boolean Matrix Multiplication (BMM) as the following: _ (AB)[i, j] = (A(i, k) ∧ B(k, j)) k ...
... Editor: Kathy Cooper Given two n × n matrices A, B over {0, 1}, we define Boolean Matrix Multiplication (BMM) as the following: _ (AB)[i, j] = (A(i, k) ∧ B(k, j)) k ...
M341 Linear Algebra, Spring 2014, Travis Schedler Review Sheet
... . Conclude that the RHS of the above does not depend on the choice of v1 and v2 (as long as (v1 , v2 ) is an orthogonal basis of V ). Hint: recall the formula for projV : Fn → V from our Gram-Schmidt orthogonalization, which we also defined to be the unique linear map such that projV |V = I|V and pr ...
... . Conclude that the RHS of the above does not depend on the choice of v1 and v2 (as long as (v1 , v2 ) is an orthogonal basis of V ). Hint: recall the formula for projV : Fn → V from our Gram-Schmidt orthogonalization, which we also defined to be the unique linear map such that projV |V = I|V and pr ...
Solutions to Homework Set 6
... 1) A group is simple if it has no nontrivial proper normal subgroups. Let G be a simple group of order 168. How many elements of order 7 are there in G? Solution: Observe that 168 = 23 · 3 · 7. Every element of order 7 generates a cyclic group of order 7 so let us count the number of such subgroups: ...
... 1) A group is simple if it has no nontrivial proper normal subgroups. Let G be a simple group of order 168. How many elements of order 7 are there in G? Solution: Observe that 168 = 23 · 3 · 7. Every element of order 7 generates a cyclic group of order 7 so let us count the number of such subgroups: ...
Condensation Method for Evaluating Determinants
... The determinant of A determines whether or not this system is solvable. In particular, if det A is nonzero, we know that the inverse matrix A–1 exists, and this in turn promises a unique solution to the system of linear equations represented by matrix A. Although 2 × 2 determinants can be calculated ...
... The determinant of A determines whether or not this system is solvable. In particular, if det A is nonzero, we know that the inverse matrix A–1 exists, and this in turn promises a unique solution to the system of linear equations represented by matrix A. Although 2 × 2 determinants can be calculated ...
D Linear Algebra: Determinants, Inverses, Rank
... this solution into Ax = y and noting that AZ vanishes. The components x p and xh are called the particular and homogeneous portions respectively, of the total solution x. (The terminology: homogeneous solution and particular solution, are often used.) If y = 0 only the homogeneous portion remains. I ...
... this solution into Ax = y and noting that AZ vanishes. The components x p and xh are called the particular and homogeneous portions respectively, of the total solution x. (The terminology: homogeneous solution and particular solution, are often used.) If y = 0 only the homogeneous portion remains. I ...
Math 2270 - Lecture 16: The Complete Solution to Ax = b
... Note that all matrices with full column rank are “tall and thin”. Now let’s take a look at the other type of rectangular matrix. Namely, one with at least as many columns as rows. Such a matrix is referred to as “short and wide” in the textbook. Suppose further than the rank of the matrix is the sam ...
... Note that all matrices with full column rank are “tall and thin”. Now let’s take a look at the other type of rectangular matrix. Namely, one with at least as many columns as rows. Such a matrix is referred to as “short and wide” in the textbook. Suppose further than the rank of the matrix is the sam ...
Linear Algebra
... x in R , the vector T(x) in R is called the image of x (under the action of T). The set of all images T(x) is called the range of T n ...
... x in R , the vector T(x) in R is called the image of x (under the action of T). The set of all images T(x) is called the range of T n ...
Real Symmetric Matrices
... 3. If A ∈ Mn (C), then the trace of the product A∗ A is the sum of all the entries of A, each multiplied by its own complex conjugate (check this). This is a non-negative real number and it is zero only if A = 0. In particular, if A ∈ Mn (R), then trace(AT A) is the sum of the squares of all the ent ...
... 3. If A ∈ Mn (C), then the trace of the product A∗ A is the sum of all the entries of A, each multiplied by its own complex conjugate (check this). This is a non-negative real number and it is zero only if A = 0. In particular, if A ∈ Mn (R), then trace(AT A) is the sum of the squares of all the ent ...
FAMILIES OF SIMPLE GROUPS Today we showed that the groups
... Today we showed that the groups An , n ≥ 6, are all simple. All but finitely many of the other finite simple groups also fall into infinite families, and these families generally consist of invertible matrices over finite fields such as Fp (the integers mod p, p a prime). Later in the course we will ...
... Today we showed that the groups An , n ≥ 6, are all simple. All but finitely many of the other finite simple groups also fall into infinite families, and these families generally consist of invertible matrices over finite fields such as Fp (the integers mod p, p a prime). Later in the course we will ...