Standards.IsMx History

Hide minor edits - Show changes to markup - Cancel

September 16, 2008, at 01:52 PM by LFS -
Changed line 298 from:

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.

to:

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.

September 16, 2008, at 01:31 PM by LFS -
Changed lines 286-287 from:
to:
September 16, 2008, at 01:25 PM by LFS -
Changed line 116 from:

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

to:

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

September 16, 2008, at 01:19 PM by LFS -
Changed lines 256-257 from:

[ width=250]Intl Standards for Matrix Algebra
[]CA - California Standards for Linear Algebra

to:

[ width=250]Intl Standards for Matrix Algebra
[]CA - California Standards for Linear Algebra

September 16, 2008, at 01:18 PM by LFS -
Changed line 11 from:

[]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.

to:

[]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.

September 16, 2008, at 01:16 PM by LFS -
Changed line 166 from:
to:
Changed line 252 from:
to:
Changed line 256 from:

[ width=250]International Standards for Matrix Algebra

to:

[ width=250]Intl Standards for Matrix Algebra

September 16, 2008, at 01:15 PM by LFS -
Changed lines 252-253 from:
to:
Changed lines 256-257 from:

[]Intl
[]CA

to:

[ width=250]International Standards for Matrix Algebra
[]CA - California Standards for Linear Algebra

Changed lines 268-271 from:
to:
Changed lines 271-274 from:
to:
Changed lines 274-275 from:
to:
Changed lines 283-284 from:
to:
Added line 300:

[tableend]

September 16, 2008, at 01:11 PM by LFS -
Changed lines 281-282 from:

[]Not specifically matrix algebra.

to:
Changed line 288 from:

[]Not specifically matrix algebra.

to:
Changed line 306 from:

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.

to:

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.

September 16, 2008, at 01:01 PM by LFS -
Changed line 113 from:

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

to:

[]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.

Changed line 116 from:

[]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.

to:

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

Added line 162:
Changed lines 250-253 from:

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


[table width=825]

to:

[table border=1 cellpadding=3 ]

Changed lines 252-261 from:
to:
Added lines 304-325:
September 16, 2008, at 11:23 AM by LFS -
Changed lines 232-233 from:
I       L   O   V   E       M   O   N   I   C   A
5  0 -6   8  -11  3   0   7   8   -7   5   2   1
to:
I       L   O   V   E       M   O   N   I   C   A
5  0  -6   8  -11  3   0   7   8   -7   5   2   1
September 16, 2008, at 11:22 AM by LFS -
Changed lines 189-191 from:

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) .

to:

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) .

Deleted lines 200-203:

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) .

Deleted line 224:

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

Changed lines 226-227 from:
I       L   O   V   E       M   O   N   I   C   A
5  0 -6   8  -11  3   0   7   8   -7   5   2   1
to:

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...

Changed lines 228-229 from:

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) $}

to:

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

Changed line 230 from:

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

to:

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

Changed lines 232-233 from:
AB = \left( {\matrix{ { - 1} & 5 & { - 1} \cr { - 2} & {11} & 7 \cr 1 & { - 5} & 2 \cr } } \right)\left( {\matrix{ 5 & 8 & 0 & { - 7} & 1 \cr 0 & { - 11} & 7 & 5 & 0 \cr { - 6} & 3 & 8 & 2 & 0 \cr } } \right) = \left( {\matrix{ 1 & { - 66} & {27} & {30} & { - 1} \cr { - 52} & { - 116} & {133} & {83} & { - 2} \cr { - 7} & {69} & { - 19} & { - 28} & 1 \cr } } \right)
to:
I       L   O   V   E       M   O   N   I   C   A
5  0 -6   8  -11  3   0   7   8   -7   5   2   1
Changed line 235 from:

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

to:

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)

Changed line 237 from:
to:

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

Changed lines 239-241 from:
to:
AB = \left( {\matrix{ { - 1} & 5 & { - 1} \cr { - 2} & {11} & 7 \cr 1 & { - 5} & 2 \cr } } \right)\left( {\matrix{ 5 & 8 & 0 & { - 7} & 1 \cr 0 & { - 11} & 7 & 5 & 0 \cr { - 6} & 3 & 8 & 2 & 0 \cr } } \right) = \left( {\matrix{ 1 & { - 66} & {27} & {30} & { - 1} \cr { - 52} & { - 116} & {133} & {83} & { - 2} \cr { - 7} & {69} & { - 19} & { - 28} & 1 \cr } } \right)
Changed line 241 from:

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

to:

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

Added lines 243-250:
September 16, 2008, at 11:17 AM by LFS -
September 16, 2008, at 11:17 AM by LFS -
Changed line 163 from:

[table border=1 cellpadding=3 width=825]

to:

[table border=1 cellpadding=3]

Changed line 165 from:
to:
Changed line 168 from:
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.
to:
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.
Changed line 186 from:
Standard MX3.0: Matrix multiplication should be introduced and motivated via a sample as below SOS Math.
to:
Standard MX3.0: Matrix multiplication should be introduced and motivated via a sample as below SOS Math.
Changed line 188 from:
to:
Changed lines 200-201 from:
to:

Changed lines 202-203 from:

[tableend]

to:
Changed line 206 from:

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

to:

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.

Added lines 208-242:
September 16, 2008, at 11:05 AM by LFS -
Changed line 191 from:

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,

to:

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) .

Changed lines 193-194 from:
{$ \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) $} .

to:

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) .

Changed line 197 from:

If we consider the following rule for matrix multiplication:

to:
Changed lines 199-200 from:
{$ \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) $} .

to:
Changed lines 204-205 from:

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:

[tableend]

Changed line 206 from:
to:

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

Deleted lines 207-215:
September 16, 2008, at 11:03 AM by LFS -
Changed lines 163-165 from:

[table width=825]

to:

[table border=1 cellpadding=3 width=825]

Changed lines 165-218 from:
to:
September 16, 2008, at 10:43 AM by LFS -
Changed lines 40-41 from:

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

to:

[]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]

Changed line 47 from:

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

to:

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

Changed lines 68-69 from:

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

to:

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

Changed line 99 from:

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

to:

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

Changed line 116 from:

[]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.

to:

[]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.

Changed line 148 from:

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.

to:

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.

September 16, 2008, at 10:13 AM by LFS -
Deleted lines 148-149:

[]7.4
[]Compute the scalar (dot) product of two vectors in n-dimensional space and know that perpendicular vectors have zero dot product.

Added lines 151-153:

[]Compute the scalar (dot) product of two vectors in n-dimensional space and know that perpendicular vectors have zero dot product.
[row]
[]7.6

Deleted line 154:
September 16, 2008, at 10:11 AM by LFS -
Changed lines 46-47 from:

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

to:

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

September 16, 2008, at 05:10 AM by LFS -
Added lines 1-176:


Page last modified on September 16, 2008, at 01:52 PM