## §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:.

ila

## 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:.

ha

## 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=b### A. 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 inR3Outline#### 1Matrices 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 inR3### Linear 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 "a#### 1:::anb

has a solution.2Indeed, there exists some collection ofnreal numbers x#### 1;:::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="a#### 1:::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 x#### 1a1+:::+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 Multiplication### Thus, 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 x#### 1a1+:::xnan2Rm

of the columns foA, where x=2 6 4x 1... x n3 7#### 5A. HavensMatrix-Vector Products and the Matrix EquationAx=b

Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3### Matrix-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 inR3### Computing 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 combination### Ax= (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 inR3### Computing 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=b### A. 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 inR3Outline#### 1Matrices 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 inR3### Linear 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 "a#### 1:::anb

has a solution.2Indeed, there exists some collection ofnreal numbers x#### 1;:::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="a#### 1:::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 x#### 1a1+:::+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 Multiplication### Thus, 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 x#### 1a1+:::xnan2Rm

of the columns foA, where x=2 6 4x 1... x n3 7#### 5A. HavensMatrix-Vector Products and the Matrix EquationAx=b

Matrices Acting on VectorsThe equationAx=bGeometry of Lines and Planes inR3### Matrix-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 inR3### Computing 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 combination### Ax= (2)2

6 412 33

7

#### 5+ (1)2

6 445 63

7

#### 5+ (1)2

6 478 93

7