Is the zero vector equal to the zero vector? Is the zero vector collinear with the zero vector? Isn't the zero vector collinear with all vectors? Why do many collinear questions have no zero vector

Is the zero vector equal to the zero vector? Is the zero vector collinear with the zero vector? Isn't the zero vector collinear with all vectors? Why do many collinear questions have no zero vector

Zero vector is a vector with arbitrary direction and zero length. It is collinear with any vector. For this reason, the answer that a vector is collinear with zero vector does not appear in the general single answer. This is generally not discussed unless the topic specifically refers to zero vector
Like the relationship between two sets, many of the answers are true subsets. Logically, true subsets are also subsets, but the answers are not subsets, but true subsets,
I hope the above answers can solve the confusion of the landlord~

, given that vector a is a non-zero vector and B is not equal to C, prove that a * b = a * C = a is perpendicular to (B-C)

From a vertical (B-C): a * (B-C) = 0A * (B-C) = 0 from a * b = a * C: a * B-A * C = 0, a * (B-C) = 0

Verification: the determinant of linear correlation (at a) of column vector group of matrix A is zero Verification: The necessary and sufficient condition for the linear correlation of M-ary vector groups A1, A2,..., an is DET (at a) = 0, where amxn = [A1, A2,..., an] At is trans (a), the transpose of A On the first floor, please describe the row 帙 of matrix A ^ t = the column 帙 of matrix A

I want to prove this proposition. Construct two homogeneous linear equations: (1) AX = 0, (2) (at a) x = 0. If the two equations have the same solution, the coefficient matrices of the two equations have the same rank, R (a) = R (at a) = n

How to use the elementary row transformation of the matrix to judge whether the vector group is linearly correlated or linearly independent?

M n-dimensional column vectors α 1, α 2,……, α m. If M > n α 1, α 2,……, α M. inevitable linear correlation
When m ≤ n, M-column matrix for n rows（ α 1, α 2,……, α m) , perform a row elementary transformation. The goal is to have R
Column. The substituent composed of the first R rows becomes an r-order identity matrix, and the whole matrix is
Fatal Frame.
If r = n., then { α 1, α 2,……, α M. linear independence
If R < n.. { α 1, α 2,……, α M. linear correlation
And: two other important problems are solved at the same time. ① the maximum irrelevant group is found
② The expression of "other" vector about this maximum independent group is found
For example（ α 1, α 2, α 3, α 4, α 5) → line elementary transformation →
2,1,3,0,0
-1,0,2,1,0
12,0.-2,0,1
0,0,0,0.0. (standard form), then:
①.｛ α 1, α 2, α 3, α 4, α 5. Linear correlation. (∵ 3 = R ＜ 4 = n)
② . the maximum independent group is { α 2, α 4, α 5. (of course not unique.)
③. α 1=2 α 2- α 4+12 α five α 3=3 α 2+2 α 4-2 α five
(the only reason for these results is that the row elementary transformation maintains the linear relationship between columns.)

If a is an M * n matrix and M > N, the row vector group of a is linearly correlated. Can you explain this sentence

It is known that R (a) < = min {m, n} = n < M
And the rank of a = the row rank of a = the column rank of A
Therefore, the row rank of a is < m (i.e. the number of row vectors)
So the row vectors of a are linearly correlated

Let a be a second-order matrix, α 1, α 2 is a linear independent 2-dimensional column vector, a α 1=0,A α 2=2 α 1+ α 2, then the non-zero eigenvalue of a is? 1 and 2 are lower footmarks

A α 1=0* α 1 = 0, so there is an eigenvalue of 0, and the corresponding eigenvector is α one
A α 2=2 α 1+ α 2. Multiply both sides by A
A^2 α 2=2A α 1+A α 2=A α two
That is (a ^ 2-A) α 2=0
Since a is a second-order matrix, it can be seen that the equation has a non-zero solution α 2. The condition is
A ^ 2-A contains characteristic value 0,
That is, set the eigenvalue as λ,λ^ 2- λ= 0, the other root is 1, and the corresponding eigenvector is α two