![](data:image/svg+xml;utf8,<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 0 72 71" width="72"></svg>)
Given the plane vector a =(2,-2), b =(3,4), a×b=a×c, then the minimum value of |c| is A2B root 2C1/2D root 2/2
A×b=a×c=-2, then -2=a*c|c|=(a*c)/|a|=-root 2/(2cosa), when cosa takes the minimum value -1,|c| gets the maximum value, select D
A×b=a×c=-2, then -2=a*c|c|=(a*c)/|a|=-root 2/(2cosa), when cosa takes the minimum value -1,|c| takes the maximum value D
It is proved that there exists n-dimensional vector x such that Ax=Bx It is proved that there is n-dimensional vector x such that Ax=Bx
The title should be a nonzero vector x.
By BC=0, r (B)+r (C)
Let A be a matrix of order n,α be a vector of order n, if A^n-1α=0, and A^nα=0, it is proved that α, Aα,..., A^n-1α are linearly independent.
Let k0 k1A...+k (n-1) A^(n-1)α=0
Simultaneous left multiplication A^(n-1)
Since A^nα=0, A^(i)α=0(i >=n)
Then k0A^(n-1)α=0 and A^n-1α=0, then k0=0
Then k1A...+k (n-1) A^(n-1)α=0
At the same time, multiply A^(n-2) by A^(n-2) in the same way to obtain k1=0.
By analogy, ki=0(i=0,1,2,3,...,n -1)
Then we get α, Aα,..., A^n-1α linearly independent
(Aα) Let A be a matrix of order n and α be a vector of dimension n. If the rank r (αT0)=r (A), then the system of linear equations () αT is the transposition of α A. Ax=α must have infinite solution B.Ax=α must have unique solution C.(Aα)(x) (αT0)(y)=0 Zero solution only D.(Aα)(x) (αT0)(y)=0 must have a non-zero solution (Aα) Let A be a square matrix of order n and α be a vector of dimension n. If the rank r (αT0)=r (A), then the system of linear equations () αT is the transposition of α A. Ax=α must have infinite solution B.Ax=α must have a unique solution C.(Aα)(x) (αT0)(y)=0 Zero solution only D.(Aα)(x) (αT0)(y)=0 must have a non-zero solution (Aα) Let A be a square matrix of order n and α be a vector of dimension n. If the rank r (αT0)=r (A), then the system of linear equations () αT is the transposition of α A. Ax=α must have infinite solution B.Ax=α must have unique solution C.(Aα)(x) (αT0)(y)=0 Zero solution only D.(Aα)(x) (αT0)(y)=0 must have a non-zero solution
R [A,α;α T,0]= r (A)
Let A, B be n-dimensional column vectors, then the rank of n-order matrix c=ab^t is r (a)=, why not equal to n, the answer is 0 or 1
Let A =(a1, a2,.an)^T, B =(b1, b2,.bn)^T
Then AB^T = a1b1 a1b2 a1b3.a1bn
a2b1 a2b2 a2b3 .a2bn
..
anb1 anb2 anb3 .anbn
Note any 2*2 submatrix aibj aibk
Asbj asbk
Its determinant is 0, so any k (or 2 or more) level of the sub-expression is equal to 0.
So rank of AB^T
A, b are all three-dimensional column vectors, matrix A = baT (b times (a transpose)), what is the rank of matrix A? Why? What if it's extended to n-dimension?
The rank of matrix A is less than or equal to 1.
Because r (A)= r (ba^T)
RELATED INFORMATIONS
- 1. Let a, b be n-dimensional nonzero column vectors, matrix A=2E (N-dimensional)-abt (transposed vector of b) What is the transposition vector of a multiplied by b if the square of A = A +2E
- 2. Matrix proof: If the matrix of order n satisfies AA^T=E, it is proved that x^TAx=0. If the matrix of order n satisfies A^T=-A, it is proved that x^TAx=0.
- 3. Let A be a real matrix of m*n and A^TA be a positive definite matrix. It is proved by data that r (A^TA)≤r (n)≤n, but I don't know where this formula comes from, and the formula is r (AB)≤r (A), r (AB)≤r (B),. I've found the problem in the book, and now there's another question: Shouldn't r (A)≤min {m, n}? Why is the formula directly r (n)≤n? Somebody help me.
- 4. Let A be the s×m matrix, B be the m×n matrix, and X be the vector of n-dimensional unknown columns. Let A be the s×m matrix, B be the m×n matrix, and X be the vector of n-dimensional unknown columns. AB has the same rank as B, i.e. r (AB)=r (B) Let A be the s×m matrix, B be the m×n matrix, and X be the vector of n-dimensional unknown columns. Let A be the s×m matrix, B be the m×n matrix, and X be the vector of n-dimensional unknown columns. AB has the same rank as B, i.e. r (AB)= r (B)
- 5. It is proved that in n-dimensional vector space, if α1.α2...αn is linearly independent, then any vector β can be linearly represented by α1.α2..αn. It is proved that any vector β can be linearly represented by α1.α2...α n if α1.α2...
- 6. N +1 n-dimensional vector, composed of vector group dimension linear (? ) Vector Group N +1 n-dimensional vector, the vector group composed of linear (? ) Vector Group
- 7. For the vector a, b, c and the real number λ, is the true proposition () in the following propositions possible to analyze these four in detail? A. If a*b=0, then a=0 or b=0 B. If λa=0, then λ=0 or a=0 C. If a^2= b^2, then a = b or a =-b D. If a*b=a*c, then b=c For the vector a, b, c, and real number λ, could the following real proposition () analyze these four in detail? A. If A*B=0, then A=0 or B=0 B. If λa=0, then λ=0 or a=0 C. If A2= B2, then A = B or A = B D. If a*b=a*c, then b=c
- 8. Is a linear algebra problem," a set of nonzero n-dimensional vectors, if they are orthogonal to each other, then they are called orthogonal vector sets "orthogonal to any two vectors? Linear algebra problem," a set of nonzero n-dimensional vectors, if they are orthogonal to each other, it is called orthogonal vector set "is any two vectors orthogonal? Linear algebra problem," a set of nonzero n-dimensional vectors, if they are orthogonal to each other, then it is called orthogonal vector set "Is any two vectors orthogonal?
- 9. Vector midpoint proof In the known isosceles triangle ABC,∠C=90°, D is the midpoint of CB, E is the point on AB, and AE=2EB.
- 10. Let the sum of the elements of any row of matrix A of order n be a and prove that a is a eigenvalue of matrix A to find the eigenvector corresponding to a Let a be the sum of the elements of any matrix A of order n be a and prove that a is a eigenvalue of matrix A to find the eigenvector corresponding to a
- 11. Let X be the normal vector of X plane through three points A (1,2,3) B (2,0,-1) C (3,-2,0) Given X be the normal vector of the X plane through three points A (1,2,3) B (2,0,-1) C (3,-2,0) Given the plane X through three points A (1,2,3) B (2,0,-1) C (3,-2,0) to find the normal vector of X-plane
- 12. The known triangle ABC is an acute triangle, and the three internal angles are AB C The known vector p=(2-2sinA,cosA + sin A) q=(1+ sin A. cos A-sin A) If p is perpendicular q, find the magnitude of the internal angle A
- 13. If A, B are the two internal angles of the acute triangle ABC... a mathematical solution ~~ 1. If A and B are two internal angles of acute triangle ABC, in which quadrant is the point P (cosB-sinA, sinB-cosA)? Why? 2. Given that the coordinate of the point P on the final edge of the acute angle α is (2sin2,-2cos2), then α is equal to? 3. Given f (sinx)= sin (4n+1) x, find f (cosx)=? Three questions for answers
- 14. If two vectors are added =0, then is the sum of their abscissa equal to 0?
- 15. On the Proof of Vector Coplanarity I didn't seem to remember all the notes the teacher wrote: Vector AC=λ1 vector AB 2 vector AD has no specific requirements for λ1λ2 Which points are coplanar? What is the relationship between point A and BCD?
- 16. The usage vector proves two parallel exercises An exercise in which the usage vector proves two sides parallel
- 17. How to prove that a vector is the normal vector of a plane
- 18. What is the projection of a on b when the vector ab is parallel and reversed What is the projection of a on b when the vector ab is parallel and opposite
- 19. 1. If a and b are collinear, b and c are collinear? 2. If the module of vector a+b is equal to the module of vector a-b, is a multiplied by b=0? 1. If a and b are collinear, then a and c are collinear? 2. If the module of vector a+b is equal to the module of vector a-b, is a multiplied by b=0?
- 20. △The lengths of the opposite sides of the three internal angles A, B and C of ABC are a, b and c respectively. Let the vector P=(a+c, b), Q=(b−a, c−a), if P∥ Q, the size of angle C is ______.