§1.4 Matrix Equation Ax = b: Linear Combination (I)
b2 b3.. . Question: For what values of b1b2
week
18.06 Problem Set 1 Solutions
Feb 11 2010 If E21 subtracts row 1 from row. 2
pset s soln
Math 2331 – Linear Algebra - 1.4 The Matrix Equation Ax=b
1.4 The Matrix Equation Ax = b. Definition Theorem Span Rm. Matrix-Vector Multiplication: Examples. Example 1 −4. 3. 2.
sec
2.5 Inverse Matrices
Elimination solves Ax D b without explicitly using the matrix A. 1 . Note 2 Find the inverses (directly or from the 2 by 2 formula) of A;B;C:.
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Now let's show that V ar(aX + b) = a 2V ar(X). This is for a b
Same kind of idea works but just want to remember this. V ar(aX + b) = E((aX + b)2) − (E(aX + b))2. =
prob
Matrix-Vector Products and the Matrix Equation Ax= b
Jan 31 2018 has a solution. 2. Indeed
Lecture
Chapter 2 - Matrices and Linear Algebra
Ax = b. In this way we see that with ci (A) denoting the ith column of A the system is expressible as x1c1 (A) + ··· + xncn (A) = b. From this equation it
chapter
The Matrix Equation Ax = b Section 1.5: Solution Sets of Linear
This section is about solving the “matrix equation” Ax = b where A is an m Exercise 2 (1.7.1): Check if the following vectors are linearly independent:.
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Table of Integrals
u(x)v (x)dx = u(x)v(x) v(x)u (x)dx. RATIONAL FUNCTIONS. (5). 1 ax + b dx = 1 a ln(ax + b). (6). 1. (x + a)2 dx = 1 x + a. (7). (x + a)n dx = (x + a)n.
IntegralTable
Math 215 HW #4 Solutions
Execute the six steps following equation (6) to findthee column space and nullspace of A and the solution to Ax = b: A =.. 1 1 2 2. 2 5 7 6. 2 3 5 2.
hw solutions
Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3Matrix-Vector Products and the Matrix Equation
Ax=bA. Havens
Department of Mathematics
University of Massachusetts, Amherst
January 31, 2018
A. HavensMatrix-Vector Products and the Matrix EquationAx=b Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3Outline1Matrices Acting on Vectors
Linear Combinations and Systems
Matrix-Vector Products
Computing Matrix-Vector Products
2The equationAx=bReturning to Systems
Some Examples in three dimensions
3Geometry of Lines and Planes inR3Vector description of a line
Planes, Displacement Vectors, and Normals
A. HavensMatrix-Vector Products and the Matrix EquationAx=b Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3Linear Combinations and SystemsA Recollection
Fix a collectiona1;:::;anof vectors inRm.1We can connect the question of whether a vectorb2Rmis a linear combination of the vectorsa1;:::;anto the question of whether the system with augmented matrix "a1:::anb
has a solution.2Indeed, there exists some collection ofnreal numbers x1;:::xnsuch that
b=x1a1+:::xnan if and only if there is a solution (x1;:::xn) to the system with the above augmented matrix. A. HavensMatrix-Vector Products and the Matrix EquationAx=b Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3 Linear Combinations and SystemsTranslating from systems to vector equations In particular, ifbis a linear combination of the columnsa1;:::;an then it must be the case that there is some matrixA0that is row-equivalent to the matrixA="a1:::ansuch that
RREF"a
1:::anb
="A0x
where x=2 6 4x 1... x n3 7 5: Conversely, if you can row reduce the augmented matrix of a system to obtain a solution, then you've realized the column vector of constantsbas a linear combination of the columns of the coecient matrixA.A. HavensMatrix-Vector Products and the Matrix EquationAx=b Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3 Matrix-Vector ProductsDening Matrix-vector Multiplication The perspective above suggests that given anmnmatrix and a vectorx2Rn, there is a natural way to create a linear combination x1a1+:::+xnan2Rmusing the columnsa1;:::;anofA.A. HavensMatrix-Vector Products and the Matrix EquationAx=b
Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3 Matrix-Vector ProductsDening Matrix-vector MultiplicationThus, we make the following denition:
Denition
Given anmnmatrixA="a
1:::anand a vectorx2Rn
we dene the matrix vector productAxto be the vector giving the linear combination x1a1+:::xnan2Rm
of the columns foA, where x=2 6 4x 1... x n3 75A. HavensMatrix-Vector Products and the Matrix EquationAx=b
Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3Matrix-Vector ProductsObservation
Matrix-vector products are only dened when the sizes of the matrix and vector are compatible { the number of components of the vectorxmust equal the number ofcolumnsof the matrix. The result will be a vector with as many components as the number of rowsof the matrix.Remark Later, we will interpret matrix-vector products as describing a special kind of transformation, called alinear transformation. In particular, anmnmatrix acts on ann-vectorx2Rnto produce anm-vectorAx2Rm, so we can describe a certain kind of map of vectors fromRntoRm. Ifn=m, these linear transformations allow us to describe geometric transformations of space, such as rotations and re ections, as well as other more general maps of n-vectors.A. HavensMatrix-Vector Products and the Matrix EquationAx=b Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3Computing Matrix-Vector ProductsAn Example
Example
Consider the matrix
A=2 6 414 72 58 36 93
7 5 and the vector x=2 6 42
1 13 7 5: ComputeAx.A. HavensMatrix-Vector Products and the Matrix EquationAx=b Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3
Computing Matrix-Vector ProductsAn Example
Example
The result of the matrix vector productAxis the linear combinationAx= (2)2
6 412 33
7
5+ (1)2
6 445 63
7
5+ (1)2
6 478 93
7
5A. HavensMatrix-Vector Products and the Matrix EquationAx=b
Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3Computing Matrix-Vector ProductsAn Example
Example
By properties of scaling and vector addition:
Ax=2Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3Matrix-Vector Products and the Matrix Equation
Ax=bA. Havens
Department of Mathematics
University of Massachusetts, Amherst
January 31, 2018
A. HavensMatrix-Vector Products and the Matrix EquationAx=b Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3Outline1Matrices Acting on Vectors
Linear Combinations and Systems
Matrix-Vector Products
Computing Matrix-Vector Products
2The equationAx=bReturning to Systems
Some Examples in three dimensions
3Geometry of Lines and Planes inR3Vector description of a line
Planes, Displacement Vectors, and Normals
A. HavensMatrix-Vector Products and the Matrix EquationAx=b Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3Linear Combinations and SystemsA Recollection
Fix a collectiona1;:::;anof vectors inRm.1We can connect the question of whether a vectorb2Rmis a linear combination of the vectorsa1;:::;anto the question of whether the system with augmented matrix "a1:::anb
has a solution.2Indeed, there exists some collection ofnreal numbers x1;:::xnsuch that
b=x1a1+:::xnan if and only if there is a solution (x1;:::xn) to the system with the above augmented matrix. A. HavensMatrix-Vector Products and the Matrix EquationAx=b Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3 Linear Combinations and SystemsTranslating from systems to vector equations In particular, ifbis a linear combination of the columnsa1;:::;an then it must be the case that there is some matrixA0that is row-equivalent to the matrixA="a1:::ansuch that
RREF"a
1:::anb
="A0x
where x=2 6 4x 1... x n3 7 5: Conversely, if you can row reduce the augmented matrix of a system to obtain a solution, then you've realized the column vector of constantsbas a linear combination of the columns of the coecient matrixA.A. HavensMatrix-Vector Products and the Matrix EquationAx=b Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3 Matrix-Vector ProductsDening Matrix-vector Multiplication The perspective above suggests that given anmnmatrix and a vectorx2Rn, there is a natural way to create a linear combination x1a1+:::+xnan2Rmusing the columnsa1;:::;anofA.A. HavensMatrix-Vector Products and the Matrix EquationAx=b
Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3 Matrix-Vector ProductsDening Matrix-vector MultiplicationThus, we make the following denition:
Denition
Given anmnmatrixA="a
1:::anand a vectorx2Rn
we dene the matrix vector productAxto be the vector giving the linear combination x1a1+:::xnan2Rm
of the columns foA, where x=2 6 4x 1... x n3 75A. HavensMatrix-Vector Products and the Matrix EquationAx=b
Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3Matrix-Vector ProductsObservation
Matrix-vector products are only dened when the sizes of the matrix and vector are compatible { the number of components of the vectorxmust equal the number ofcolumnsof the matrix. The result will be a vector with as many components as the number of rowsof the matrix.Remark Later, we will interpret matrix-vector products as describing a special kind of transformation, called alinear transformation. In particular, anmnmatrix acts on ann-vectorx2Rnto produce anm-vectorAx2Rm, so we can describe a certain kind of map of vectors fromRntoRm. Ifn=m, these linear transformations allow us to describe geometric transformations of space, such as rotations and re ections, as well as other more general maps of n-vectors.A. HavensMatrix-Vector Products and the Matrix EquationAx=b Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3Computing Matrix-Vector ProductsAn Example
Example
Consider the matrix
A=2 6 414 72 58 36 93
7 5 and the vector x=2 6 42
1 13 7 5: ComputeAx.A. HavensMatrix-Vector Products and the Matrix EquationAx=b Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3
Computing Matrix-Vector ProductsAn Example
Example
The result of the matrix vector productAxis the linear combinationAx= (2)2
6 412 33
7
5+ (1)2
6 445 63
7
5+ (1)2
6 478 93
7