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Show that eigen vectors corresponding to distinct eigen values of matrix A are linearly independent?
Let l1,...,lr be distinct eigenvalues of an nxn matrix A, and let v1,...,vr be the corresponding eigenvectors. This proof is by induction:
The case r=1 is obvious since k1 is not the zero vector.
Suppose that the theorem holds true for r=m-1, i.e. {k1,...,km-1} is linearly independent. This means
c1k1+c2k2+...+cm-1km-1=0 (1)
iff ci=0 for all i. Now suppose that
a1k1+a2k2+...+am-1km-1+amkm=0 (2)
If am is not equal to 0, then
km=b1k1+b2k2+...+bm-1km-1 (3)
where bi=-ai/am. Multiplying both sides of (3) by lmI-A (where I is the nxn identity matrix) and using the fact that Aki=liki, we have:
0=(l1-lm)b1k1+(l2-lm)b2k2+...+(lm-1-lm)bm-1km-1.
By our induction hypothesis we have that (li-lm)bi=0 for each i. Thus by (3) km is the zero vector, which is a contradiction since it is an eigenvector. Thus it must be that am is zero. If am=0 in (2), then all of the ai=0 by the induction hypothesis. Thus {k1,...,km} is linearly independent. By induction this is true for all 1<=r<=n.
What else can I help you with?
If a set of vectors spans R3 then the set is linearly independent?
No it is not. It's possible to have to have a set of vectors that are linearly dependent but still Span R^3. Same holds true for reverse. Linear Independence does not guarantee Span R^3. IF both conditions are met then that set of vectors is called the Basis for R^3. So, for a set of vectors, S, to be a Basis it must be:(1) Linearly Independent(2) Span S = R^3.This means that both conditions are independent.
1 1 Check whether the following set of vectors is LD or LI?
(i) They are linearly dependent since the 2nd vector is twice the 1st vector. All 3 vectors lie in the x-z plane, so they don't span 3D space. (ii) They are linearly independent. Note that the cross-product of the first two is (-1,1,1). If the third vector is not perpendicular to the above cross-product, then the third vector does not lie in the plane defined by the first two vectors. (-1,1,1) "dot" (1,1,-1) = -1+1-1 = -1, not zero, so 3rd vector is not perpendicular to the cross product of the other two.
When are vectors said to be perpendicular?
Perpendicular means that the angle between the two vectors is 90 degrees - a right angle. If you have the vectors as components, just take the dot product - if the dot product is zero, that means either that the vectors are perpendicular, or that one of the vectors has a magnitude of zero.
When are two vectors identical?
Two vectors are identical when all their components are identical. An alternative definition, for vectors used in physics, is that they are identical when both the magnitude and the direction are identical.
Can the directions of the sum of two two vectors be equal to the directions of difference of two vectors?
Yes.
Show that only N orthogonal vectors can be formed from N linearly independent vectors?
shut up now
If a set of vectors spans R3 then the set is linearly independent?
No it is not. It's possible to have to have a set of vectors that are linearly dependent but still Span R^3. Same holds true for reverse. Linear Independence does not guarantee Span R^3. IF both conditions are met then that set of vectors is called the Basis for R^3. So, for a set of vectors, S, to be a Basis it must be:(1) Linearly Independent(2) Span S = R^3.This means that both conditions are independent.
How do you find a basis for a vector space?
To find a basis for a vector space, you need to find a set of linearly independent vectors that span the entire space. One approach is to start with the given vectors and use techniques like Gaussian elimination or solving systems of linear equations to determine which vectors are linearly independent. Repeating this process until you have enough linearly independent vectors will give you a basis for the vector space.
What are non proportional vectors?
Non-proportional vectors are vectors that do not have a constant scalar multiple relationship between them. In other words, they do not lie on the same line or in the same direction. Non-proportional vectors are linearly independent and have different magnitudes and directions.
How do you find the dimension of the subspace of R4 consisting of the vectors a plus 2b plus c b-2c 2a plus 2b plus c 3a plus 5b plus c?
The dimension of a space is defined as the number of vectors in its basis. Assuming your vectors are 1,2,1,0 0,1,-2,0 2,2,1,0 and 3,5,1,0 (extra zeros because you are in R4) then you must first check to see if they are linearly indepent. If all the vectors are linearly independent then the subspace defined by those vectors has a dimension 4, as there are 4 vectors in the basis.
What is the term given to vectors that dont lie in a straight line but instead point in different directions?
The term for vectors pointing in different directions is called linearly independent vectors. These vectors do not lie on the same line or plane, and they provide unique information to describe a space.
How can you tell if a matrix is invertible?
An easy exclusion criterion is a matrix that is not nxn. Only a square matrices are invertible (have an inverse). For the matrix to be invertible, the vectors (as columns) must be linearly independent. In other words, you have to check that for an nxn matrix given by {v1 v2 v3 ••• vn} with n vectors with n components, there are not constants (a, b, c, etc) not all zero such that av1 + bv2 + cv3 + ••• + kvn = 0 (meaning only the trivial solution of a=b=c=k=0 works).So all you're doing is making sure that the vectors of your matrix are linearly independent. The matrix is invertible if and only if the vectors are linearly independent. Making sure the only solution is the trivial case can be quite involved, and you don't want to do this for large matrices. Therefore, an alternative method is to just make sure the determinant is not 0. Remember that the vectors of a matrix "A" are linearly independent if and only if detA�0, and by the same token, a matrix "A" is invertible if and only if detA�0.
What is rational dimension?
Rational dimension refers to the dimension of a vector space over the field of rational numbers. It is the minimum number of linearly independent vectors needed to span the entire vector space. The rational dimension can differ from the ordinary dimension of a vector space if the vectors are over a field other than the rational numbers.
What does the addition of 2 vectors give you?
Adding two vectors results in a new vector that represents the combination of the two original vectors. The new vector is defined by finding the sum of the corresponding components of the two vectors.
1 1 Check whether the following set of vectors is LD or LI?
(i) They are linearly dependent since the 2nd vector is twice the 1st vector. All 3 vectors lie in the x-z plane, so they don't span 3D space. (ii) They are linearly independent. Note that the cross-product of the first two is (-1,1,1). If the third vector is not perpendicular to the above cross-product, then the third vector does not lie in the plane defined by the first two vectors. (-1,1,1) "dot" (1,1,-1) = -1+1-1 = -1, not zero, so 3rd vector is not perpendicular to the cross product of the other two.
What is meant by orthogonal directions of polarization?
Orthogonal directions of polarization refer to two perpendicular directions in which an electromagnetic wave's electric field oscillates. In these directions, the electric fields are independent of each other and can be represented as perpendicular vectors. This property is commonly seen in linearly polarized light.
Is it possible to add any 2 vectors?
Yes, it is possible to add any two vectors as long as they have the same number of dimensions. The result of adding two vectors is a new vector whose components are the sum of the corresponding components of the original vectors.
