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Linear Algebra
Linear algebra is the detailed study of vector spaces. With applications in such disparate fields as sociology, economics, computer programming, chemistry, and physics, including its essential role in mathematically describing quantum mechanics and the theory of relativity, linear algebra has become one of the most essential mathematical disciplines for the modern world. Please direct all questions regarding matrices, determinants, eigenvalues, eigenvectors, and linear transformations into this category.
2,176 Questions
I cannot be sure without knowing the context of your question. However, it might be that you are referring to the power series that defines the exponential function. It's difficult to write it down here because there is no provision for writing mathematical formulae.
Please see the link.
How do you show that a square matrix A is similar to its transpose?
First we will handle the diagonalizable case.
Assume A is diagonalizable, A=VDV-1.
Thus AT=(V-1)TDVT,
and D= VT AT(V-1)T.
Finally we have that A= VVT AT(V-1)TV-1, hence A is similar to AT
with matrix VVT.
If A is not diagonalizable, then we must consider its Jordan canonical form,
A=VJV-1, where J is block diagonal with Jordan blocks along the diagonal.
Recall that a Jordan block of size m with eigenvalue at L is a mxm matrix having L along the diagonal and ones along the superdiagonal.
A Jordan block is similar to its transpose via the permutation that has ones along the antidiagonal, and zeros elsewhere.
With this in mind we proceed as in the diagonalizable case,
AT=(V-1)TJTVT.
There exists a block diagonal permutation matrix P such that
JT=PJPT, thus J=PTVT AT(V-1)TP.
Finally we have that A= VPTVT AT(V-1)TPV-1, hence A is similar to AT
with matrix VPTVT.
Q.E.D.
A fundamental unit is a unit that cannot be broken down any further. Other units are based on derivatives of these units.
http://en.wikipedia.org/wiki/Fundamental_unit
In most countries, the SI base unit system is used
http://en.wikipedia.org/wiki/SI_base_unit
Things like distance and time cannot be broken down further, so base units are used (metre and second respectively).
An example of a derived unit is km/h (kilometres per hour, often seen kmph), where kilometres are based on 1000 metres, and hours being 3600 seconds (60 x 60), where speed is equal to distance divided by time.
What does slope intercept look like?
The standard form of the slope-intercept equation is:
y = mx + b
where "m" is the slope, and "b" is the y-intercept.
Program on matrix multiplication?
#include<stdio.h>
void main()
{
int a[3][3],b[3][3],c[3][3],i,j,k;
clrscr();
printf("Enter elements of A:");
for(i=0;i<=2;i++)
for(j=0;j<=2;j++)
scanf("%d",&a[i][j]);
printf("Enter elements of B:");
for(i=0;i<=2;i++)
for(j=0;j<=2;j++)
scanf("%d",&b[i][j]);
printf("A:");
for(i=0;i<=2;i++)
{
for(j=0;j<=2;j++)
printf("%d ",a[i][j]);
printf("");
//To change line.
}
printf("B:");
for(i=0;i<=2;i++)
{
for(j=0;j<=2;j++)
printf("%d ",b[i][j]);
printf("");
}
k=0;
while(k<=2)
{
for(i=0;i<=2;i++)
{
int sum=0;
for(j=0;j<=2;j++)
sum=sum+a[i][j]*b[j][k];
c[i][k]=sum;
}
k++;
}
printf("Result: ");
for(i=0;i<=2;i++)
{
for(j=0;j<=2;j++)
printf("%d ",c[i][j]);
printf("");
}
getch();
}
What are the different ways of the slope of a line?
'uphill' it is positive
'downhill' it is negative
There is only one solution set. Depending on the inequalities, the set can be empty, have a finite number of solutions, or have an infinite number of solutions. In all cases, there is only one solution set.
How do you Write a C program for matrix?
/* multiplication of a 3*3 matrix*/
#include<stdio.h>
main()
{
int a[3][3],b[3][3],c[3][3];
int i,j,k;
printf("enter the elements in A matrix:\n");
for(i=0;i<=2;i++)
{
for(j=0;j<=2;j++)
{
scanf("%d",&a[i][j]);
}
}
printf("enter b matrix:\n");
for(i=0;i<=2;i++)
{
for(j=0;j<=2;j++)
{
scanf("%d",&b[i][j]);
}
}
for(i=0;i<=2;i++)
{
printf("\n");
for(j=0;j<=2;j++)
{
c[i][j]=0;
for(k=0;k<=2;k++)
{
c[i][j] = c[i][j]+a[i][k] * b[k][j];
}
}
}
printf("multiplication matrix is:\n");
for(i=0;i<=2;i++)
{
for(j=0;j<=2;j++)
{
printf("%d\t",c[i][j]);
}
printf("\n");
}
}
It is a quadratic expession and when factored is: (x-6)(x+7).
When do you call a pair of linear equation inconsistent?
When there is no ordered n-tuple (pair, triplet, quartet - depending on the number of dimensions) that satisfies both equations at the same time. In graph form, it is when the straight lines representing the two equations are non-intersecting.
In 2 dimensions non-intersection implies that the lines are parallel.
When can matrices be added or subtracted?
When they have the same arrangement of numbers.
30-3 5+2 6-1 1+2 3-1 4+10
- Yes
6-0 9-4 1+2 2•2 10-7 5+8
2-1 4+2 1-3
- No
6-2 9+1 4+2
What is the system of equations?
A system of equations is a set of equations with more than one variable dealing with the same material. If there are 2 variables, then the system must have 2 equations before it can be solved. 3 variables need 3 equations, etc.
Yes.
If you were to plot y=23 on a graph, you'd have a straight horizontal line where y=23 (because no matter what the value of x is on the graph, y is always 23).
As the line that is plotted is straight, the equation is considered linear.
It is 1.
Which is greater a rate of change of 60 miles per hour or 1 mile per hour?
A rate of change of 60 miles per hour.
Think of it in terms of a car. The car is going to go a father distance if it is going 60 miles per hour as opposed to 1 mile per hour.
How do you choose the appropriate conversion factors?
Normally, you do not choose them: you calculate them.
It reminds me nothing in particular that I can remember
What are the three different possible solutions of a linear system?
If the lines cross then there is one solution. If they are on top of each other then there are infinite solutions. If they are parallel then there are no solutions.
