In R4, find two linearly independent vectors A3 and A4 orthogonal to A1 = (1,0,1,0) t and A2 = (1,0,1,1) t, and find the standard orthogonal basis The answer unites A1 and A2. The orthonormal basis has four vectors, but the inner product of A1 and A2 is not zero. Shouldn't the four vectors be orthogonal to each other and the inner product be zero?

In R4, find two linearly independent vectors A3 and A4 orthogonal to A1 = (1,0,1,0) t and A2 = (1,0,1,1) t, and find the standard orthogonal basis The answer unites A1 and A2. The orthonormal basis has four vectors, but the inner product of A1 and A2 is not zero. Shouldn't the four vectors be orthogonal to each other and the inner product be zero?

X1 + X3 = 0.x1 + X3 + X4 = 0, A3 = (1,0, - 1,0), A4 = (1,1, - 1,0)
Orthogonalization B3 = A3. B4 = A4 - [a3a4 / A3 & sup2;] A3 = (0,1,0,0)
Orthonormal basis C3 = (1 / √ 2,0, - 1 / √ 2,0) C4 = (0,1,0,0)

A1 = [1, 2, 3], find the non-zero vectors A2, A3, so that A1, A2, A3 are orthogonal vectors

Let x = (x1, X2, x3) be orthogonal to A1, then
x1+2x2+3x3 = 0.
Taking a set of orthogonal basic solution system is the solution, which is a common method
Let x2 = 1, X3 = 0 get A1 = (- 2,1,0) ^ t -- the normal value is
Take X1 = 1, X2 = 2, and get A2 = (1,2,5 / 3) ^ T. -- the values of X1 and X2 satisfy the orthogonality with A1, and then substitute them into the equation to determine X3

Find the rank of vector group A1 = (1,1,1,1), A2 = (1,1, - 1, - 1), A3 = (1-1,1, - 1), A4 = (1, - 1, - 1,1), A5 = (1,2,1,1)

a1^T,a2^T,a3^T,a4^T,a5^T
1 1 1 1 1
1 1 -1 -1 2
1 -1 1 -1 1
1 -1 -1 1 1
R2-r1, r3-r1, r4-r1 were obtained
1 1 1 1 1
0 0 -2 -2 1
0 -2 0 -2 0
0 -2 -2 0 0
R4-r3
1 1 1 1 1
0 0 -2 -2 1
0 -2 0 -2 0
0 0 -2 2 0
R2R3
1 1 1 1 1
0 -2 0 -2 0
0 0 -2 -2 1
0 0 -2 2 0
R4-r3
1 1 1 1 1
0 -2 0 -2 0
0 0 -2 -2 1
0 0 0 4 -1
So the rank is 4

If the rank of vector group A1, A2, A3, A4, A5 is equal to that of vector group A1, A3, A5, the two vector groups ()
Must be equivalent?
Must not be equivalent?
Not necessarily equivalent?

If vector group A2, or A4 cannot be expressed linearly by vector group A1, A3, A5, then the rank of vector group A1, A2, A3, A4, A5 must be greater than that of vector group A1, A3, a5. It is contradictory to the rank of vector group A1, A2, A3, A4, A5 and vector group A1, A3, a5