## Standards.IsMx History

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Personal comment by LFS on the California Linear Algebra Standards: In my opinion, the CA standards for Linear Algebra only **appear** to cover less material than the International Standard for Matrix Algebra. In fact, the opposite is true. Each CA standard - by its very conciseness - fails to fully expose the underlying concepts and content that needs to be understood in order to accomplish its goal. My personal goal in the International Standard was to do exactly this.

Personal comment by LFS on the California Linear Algebra Standards: In my opinion, the CA standards for Linear Algebra only **appear** to cover less material than the International Standard for Matrix Algebra. In fact, the opposite is true. Each CA standard - by its very conciseness - fails to fully expose the underlying concepts and content that need to be understood in order to accomplish its goal. My personal goal in the International Standard was to do exactly this.

[]Relate properties of determinants to linear systems that are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions.

[]Can relate properties of determinants to linear systems that are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions.

[ width=250]Intl Standards for Matrix Algebra

[]CA - California Standards for Linear Algebra

[ width=250]Intl Standards for Matrix Algebra

[]CA - California Standards for Linear Algebra

[]FOCUS: Students will be encouraged to develop an understanding of matrices. They will also appreciate the diverse nature of matrix algebra and its value in solving problems.

[]FOCUS: Students will be encouraged to develop an understanding of matrices and an appreciation of the diverse nature of matrix algebra and its value in solving problems from all branches of mathematics.

[ width=250]International Standards for Matrix Algebra

[ width=250]Intl Standards for Matrix Algebra

[]Intl

[]CA

[ width=250]International Standards for Matrix Algebra

[]CA - California Standards for Linear Algebra

[tableend]

[]Not specifically matrix algebra.

[]Not specifically matrix algebra.

Personal comment by LFS on the California Linear Algebra Standards: In my opinion, the CA standards appear to cover far less material than the International Standard. In fact, the opposite is true. Each standard - by its very conciseness - fails to fully expose the concepts and content that needs to be understood in order to accomplish its goal.

Personal comment by LFS on the California Linear Algebra Standards: In my opinion, the CA standards for Linear Algebra only **appear** to cover less material than the International Standard for Matrix Algebra. In fact, the opposite is true. Each CA standard - by its very conciseness - fails to fully expose the underlying concepts and content that needs to be understood in order to accomplish its goal. My personal goal in the International Standard was to do exactly this.

[]Identify linear and non-linear systems; homogeneous and non-homogeneous systems.

[]Can state and use Cramer’s Rule for using determinants to solve linear 2 \times 2 systems and linear 3 \times 3 systems and understand how to check and/or interpret their results graphically.

[]Can state and use Cramer’s Rule for solving linear 2 \times 2 systems and linear 3 \times 3 systems and understand how to check and/or interpret their results graphically.

[]Relate properties of determinants to linear systems that are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions.

This set of standards is still under development. Please send any suggestions, comments and critisms to contact.

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Before we give the formal definition of how to multiply two matrices, we will discuss an example from a real life situation. Consider a city with two kinds of population: the inner city population and the suburb population. We assume that every year 40% of the inner city population moves to the suburbs, while 30% of the suburb population moves to the inner part of the city. Let Io be the initial population of the inner city I_o and S_o be the initial population of the suburban area. So after one year, the population of the inner part is I_1=0.6 I_o+0.3S_o , while the population of the suburbs is S_1=0.4I_o+0.7S_o . After two years, the population of the inner city is I_2 =0.6 (0.6 I_o + 0.3 S_o) + 0.3 (0.4 I_o + 0.7 S_o) and the suburban population is given by S_2 =0.4 (0.6 I_o + 0.3 S_o) + 0.7(0.4 I_o + 0.7 S_o) .

Is there a nice way of representing the two populations after a certain number of years? Let us show how matrices may be helpful to answer this question. Let us put the coefficients in one matrix and the two populations in a table (meaning a column object with two entries):

the coefficients: \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) and the populations: \left( {\matrix{ {I_o } \cr {S_o } \cr } } \right) . Then, \left( {\matrix{ {I_1 } \cr {S_1 } \cr } } \right) = \left( {\matrix{ {0.6I_o + 0.3S_o } \cr {0.4I_o + 0.7S_o } \cr } } \right) .

Before we give the formal definition of how to multiply two matrices, we will discuss an example from a real life situation. Consider a city with two kinds of population: the inner city population and the suburb population. We assume that every year 40% of the inner city population moves to the suburbs, while 30% of the suburb population moves to the inner part of the city.

Let I_o be the initial population of the inner city and S_o be the initial population of the suburban area. So after one year, the population of the inner part is I_1=0.6 I_o+0.3S_o , while the population of the suburbs is S_1=0.4I_o+0.7S_o . After two years, the population of the inner city is I_2 =0.6 (0.6 I_o + 0.3 S_o) + 0.3 (0.4 I_o + 0.7 S_o) and the suburban population is given by S_2 =0.4 (0.6 I_o + 0.3 S_o) + 0.7(0.4 I_o + 0.7 S_o) . Is there a nice way of representing the two populations after a certain number of years? Let us show how matrices may be helpful to answer this question. Let us put the coefficients in one matrix and the two populations in a table (meaning a column object with two entries): the coefficients: \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) and the populations: \left( {\matrix{ {I_o } \cr {S_o } \cr } } \right) . Then, \left( {\matrix{ {I_1 } \cr {S_1 } \cr } } \right) = \left( {\matrix{ {0.6I_o + 0.3S_o } \cr {0.4I_o + 0.7S_o } \cr } } \right) .If we consider the following rule for matrix multiplication: \left( {\matrix{ a & b \cr c & d \cr } } \right) \cdot \left( {\matrix{ I \cr S \cr } } \right) = \left( {\matrix{ {aI + bS} \cr {cI + dS} \cr } } \right) ,

we have: \left( {\matrix{ {I_1 } \cr {S_1 } \cr } } \right) = \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) \cdot \left( {\matrix{ {I_o } \cr {S_o } \cr } } \right) and \left( {\matrix{ {I_2 } \cr {S_2 } \cr } } \right) = \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) \cdot \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) \cdot \left( {\matrix{ {I_o } \cr {S_o } \cr } } \right) .

If we consider the following rule for matrix multiplication: \left( {\matrix{ a & b \cr c & d \cr } } \right) \cdot \left( {\matrix{ I \cr S \cr } } \right) = \left( {\matrix{ {aI + bS} \cr {cI + dS} \cr } } \right) .

we have: \left( {\matrix{ {I_1 } \cr {S_1 } \cr } } \right) = \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) \cdot \left( {\matrix{ {I_o } \cr {S_o } \cr } } \right) and \left( {\matrix{ {I_2 } \cr {S_2 } \cr } } \right) = \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) \cdot \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) \cdot \left( {\matrix{ {I_o } \cr {S_o } \cr } } \right) .

To every letter we will associate a number. The easiest way to do that is to associate 0 to a blank or space, 1 to A, 2 to B, etc... Another way is to associate 0 to a blank or space, 1 to A, -1 to B, 2 to C, -2 to D, etc... Let us use the second choice. So our message is given by the string

To every letter we will associate a number. The easiest way to do that is to associate 0 to a blank or space, 1 to A, 2 to B, etc...

Now we rearrange these numbers into a matrix B (with 3 rows so that we can multiply AB). For example, we have {$ B = \left( {\matrix{ 5 & 8 & 0 & { - 7} & 1 \cr

0 & { - 11} & 7 & 5 & 0 \cr { - 6} & 3 & 8 & 2 & 0 \cr } } \right) $}

Another way is to associate 0 to a blank or space, 1 to A, -1 to B, 2 to C, -2 to D, etc...

Then we perform the product AB, where A is the matrix found above. We get

Let us use the second choice. So our message is given by the string

The encrypted message to be sent is: 1, -52, -7, -66, …, -1, -2, 1

Now we rearrange these numbers into a matrix B (with 3 rows so that we can multiply AB). For example, we have B = \left( {\matrix{ 5 & 8 & 0 & { - 7} & 1 \cr 0 & { - 11} & 7 & 5 & 0 \cr { - 6} & 3 & 8 & 2 & 0 \cr } } \right)

Then we perform the product AB, where A is the matrix found above. We get

This set of standards is still under development. Please send any suggestions, comments and critisms to contact.

The encrypted message to be sent is: 1, -52, -7, -66, …, -1, -2, 1

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**Standard MX1.0:**In order to give the student an understanding of where matrices come from, the definition of a matrix should be along the lines of the following sample SOS Math.

**Standard MX3.0:**Matrix multiplication should be introduced and motivated via a sample as below SOS Math.

[tableend]

This set of standards is still under development. Please send any suggestions, comments and critisms to contact?.

There are many ways to encrypt a message. And the use of coding has become particularly significant in recent years (due to the explosion of the internet for example). One way to encrypt or code a message uses matrices and their inverse. Indeed, consider a fixed invertible matrix A. Convert the message into a matrix B such that AB is possible to perform. Send the message generated by AB. At the other end, they will need to know A^{-1} in order to decrypt or decode the message sent. Indeed, we have: A^{-1}(AB)=B , which is the original message.

the coefficients: \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) and the populations: \left( {\matrix{ {I_o } \cr {S_o } \cr } } \right) . Then,

the coefficients: \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) and the populations: \left( {\matrix{ {I_o } \cr {S_o } \cr } } \right) . Then, \left( {\matrix{ {I_1 } \cr {S_1 } \cr } } \right) = \left( {\matrix{ {0.6I_o + 0.3S_o } \cr {0.4I_o + 0.7S_o } \cr } } \right) .

} } \right) $} .

If we consider the following rule for matrix multiplication: \left( {\matrix{ a & b \cr c & d \cr } } \right) \cdot \left( {\matrix{ I \cr S \cr } } \right) = \left( {\matrix{ {aI + bS} \cr {cI + dS} \cr } } \right) .

we have: \left( {\matrix{ {I_1 } \cr {S_1 } \cr } } \right) = \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) \cdot \left( {\matrix{ {I_o } \cr {S_o } \cr } } \right) and \left( {\matrix{ {I_2 } \cr {S_2 } \cr } } \right) = \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) \cdot \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) \cdot \left( {\matrix{ {I_o } \cr {S_o } \cr } } \right) .

If we consider the following rule for matrix multiplication:

} } \right) $} .

we have: \left( {\matrix{ {I_1 } \cr {S_1 } \cr } } \right) = \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) \cdot \left( {\matrix{ {I_o } \cr {S_o } \cr } } \right) and {$ \left( {\matrix{ {I_2 } \cr {S_2 } \cr } } \right) = \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr

} } \right) \cdot \left( {\matrix{ {0.6} & {0.3} \cr {0.4} & {0.7} \cr } } \right) \cdot \left( {\matrix{ {I_o } \cr {S_o } \cr } } \right) $}.

[tableend]

This set of standards is still under development. Please send any suggestions, comments and critisms to contact?.

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[] Up one level

[]Understand the properties of matrix addition, the definition of the additive identity (zero matrix ), the additive inverse (-A), commutative and associative properties. [row]

[]Understand the properties of matrix addition, the definition of the additive identity (zero matrix O_{m,n} ), the additive inverse (-A), commutative and associative properties.

[row]

[]Understand that the transpose of an matrix is an matrix.

[]Understand that the transpose of an m \times n matrix is an n \times m matrix.

[]Know the definition of the identity matrix and that I A = A I = A.

[]Know the definition of the identity matrix I_{m,n} and that I A = A I = A .

[]Know the definition of the inverse matrix and its properties

[]Know the definition of the inverse matrix and its properties.

[]Can state and use Cramer’s Rule for solving linear systems and linear systems and understand how to check and/or interpret their results graphically.

[]Can state and use Cramer’s Rule for solving linear 2 \times 2 systems and linear 3 \times 3 systems and understand how to check and/or interpret their results graphically.

Are familiar with the geometric interpretations of the determinants of and matrices as the area and volume of the parallelepipeds spanned by the images under the matrices of the standard basis vectors in two-dimensional and three-dimensional spaces.

Are familiar with the geometric interpretations of the determinants of 2 \times 2 and 3 \times 3 matrices as the area and volume of the parallelepipeds spanned by the images under the matrices of the standard basis vectors in two-dimensional and three-dimensional spaces.

[]7.4

[]Compute the scalar (dot) product of two vectors in n-dimensional space and know that perpendicular vectors have zero dot product.

[]Compute the scalar (dot) product of two vectors in n-dimensional space and know that perpendicular vectors have zero dot product.

[row]

[]7.6

[]Understand that the transpose of an matrix is an matrix. [row]

[]Understand that the transpose of an matrix is an matrix.

[row]