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

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

│A│cos θ

Let n-dimensional vector groups a1, a2, a3 be linearly independent, and prove that vector groups a1+2a2, a2+2a3, a3+2a4 are linearly independent.

Let k1, k2, k3 be such that k1(a1+2a2)+k2(a2+2a3)+k3(a3+2a1)=0, then (k1+2k3) a1+(2k1+k2) a2+(2k2+k3) a3=0, because a1, a2, a3 are linearly independent, then k1+2k3=0, then k1+2k1+k2=02+k2+k3=0, then k1=k2=k3=0, so vector group a1+2a2, a2+2a3, a3+2a...

Linear algebra, vector, with n-dimensional vector a1, a2, a3 linear independence, prove 3a1+2a2, a2-a3,4a3-5a1 linear independence. Linear algebra, vector, to know the linear independence of n-dimensional vectors a1, a2, a3, prove the linear independence of 3a1+2a2, a2-a3,4a3-5a1.

Prove that r (a1, a2, a3)=3 because a1, a2, a3 are linearly independent
By known (3a1+2a2, a2-a3,4a3-5a1)=(a1, a2, a3) K
K=
3 0 -5
2 1 0
0 -1 4
= 22 =0
So K is a reversible matrix.
So r (3a1+2a2, a2-a3,4a3-5a1)=r ((a1, a2, a3) K)=r (a1, a2, a3)=3
So 3a1+2a2, a2-a3,4a3-5a1 are linearly independent.

Let a1,a2,a3...an be a set of n-dimensional vectors, it is proved that these n vectors are linearly independent if and only if any n... Let a1,a2,a3...an be a set of n-dimensional vectors, it is proved that any n-dimensionalctors are linearly independent if and only if all n-dimensional vectors can be linearized. Let a1,a2,a3...an be a set of n-dimensional vectors, it is proved that these n vectors are linearly independent if and only if any n... Let a1,a2,a3...an be a set of n-dimensional vectors, it is proved that these n-vectors are linearly independent if and only if all n-dimensional vectors can be linearized.

Necessary condition: any (n+1) n-dimensional vector must be linear correlation, that is, any n-dimensional vector b can be expressed by a1,a2,a3...an linear table.
Sufficient condition: obvious

The necessary condition is that any (n+1) n-dimensional be linear correlation, that is, any n-dimensional vector b can be expressed by a1,a2,a3...an linear table.
Sufficient conditions: apparent

If the vector groups α1,α2,α3,α4 are linearly independent and each of them is orthogonal to the vector β, then the vector group α1,α2,α3,α4 And β A. Certain linear correlation B. Certain linear independence C. Possible linear correlation or linear independence If the vector group α1,α2,α3,α4 is linearly independent and each of them is orthogonal to the vector β, then the vector group α1,α2,α3,α4 And β A. Certain linear correlation B. Certain linear independence C. Possible linear correlation or linear independence

Let A1α1+A2α2+A3α3+A4α4+A5β=0..(1) If both sides of the equation are multiplied by β, then A1α1β+A2α2β+A3α3β+A4α4β+A5βαi is orthogonal to β, then αiβ=0, i=1,2,3,4, then A5β=0. If β is not a zero vector, then A5=0, then (1) Formula.

Let A1α1+A2α2+A3α3+A4α4+A5β=0..(1) If both sides of the equation are multiplied by β, then A1α1β+A2α2β+A3α3β+A4α4β+A5β=0. Since αi is orthogonal to β, then αiβ=0, i=1,2,3,4, then A5β=0. If β is not a vector of 0, then A5=0, then (1).

A1, a2,...an are a set of n-dimensional vectors. It is proved that the necessary and sufficient conditions for their linear independence are that any set of n-dimensional vectors can be linearly represented by them. A1, a2,...an are a set of n-dimensional vectors. It is proved that the necessary and sufficient condition for their linear independence is that any set of n-dimensional vectors can be linearly represented by them.

Demonstrate Necessity
Let a be any n-dimensional vector
Because a1 is not a2……  An linearly independent
While a1 a2……  An a is n+1 n-dimensional vectors
Is a linearly related
So a can be …… by a1 a2 An linear representation
And the expression is unique.
The sufficiency is known that any n-dimensional vector can be …… by a1 a2  An linear representation,
Therefore, unit coordinate vector group e1 e2…… En can be …… by a1 or a2  An linear representation,
Then n=R (e1 e2…… En)≤R (a1 a2……  An)≤n
I.e. R (a1 a2…… An)= n
So a1 a2……  An linearly independent