What Is the Necessary and Sufficient Condition for Linear Independence of n-dimensional Column Vector What is the Necessary and Sufficient Condition for Linear Independence of n-dimensional Column Vector

What Is the Necessary and Sufficient Condition for Linear Independence of n-dimensional Column Vector What is the Necessary and Sufficient Condition for Linear Independence of n-dimensional Column Vector

There are several formulations. I say only two:
The necessary and sufficient condition for linear independence of m n-dimensional column vectors is that there is no one vector in the m n-dimensional column vectors, which can be linearly represented by the other vectors.
The necessary and sufficient condition for linear independence of m n-dimensional column vectors is that there is no set of corresponding coefficients which are not all zero, so that the m n-dimensional column vectors are n-dimensional zero vectors after multiplication and addition of corresponding coefficients.

What is the geometric meaning of the n-dimensional vector

It's very simple. Just because we are in three-dimensional space, it's not easy to perceive a measure larger than three-dimensional. Let's start with three-dimensional. For example, the vector {x1, x2, x3} in three-dimensional space must be decomposed into {x1, x2, x3}=x1{1,0,0}+x2{0,1,0}+x3{0,0,1}. The three components {1,0,0}{0,1,0}{0,0,1} are linearly independent. And...

Let vector a be a column vector of n dimension, a^t*a=1, let H=E-2a*a^t, and prove that H is an orthogonal matrix. (E-2aa^T)^T How? Let vector a be a column vector of n dimension, a^t*a=1, let H=E-2a*a^t, and prove that H is an orthogonal matrix. (E-2aa^T)^T How to find?

H^T=(E-2aa^T)^T
=E^T—2(a^T)^T·a^T
=E-2aa^T


H·H^T=(E-2aa^T)·(E-2aa^T)
=E-2aa^T-2aa^T+4aa^T·aa^T
=E-4aa^T+4a·(a^T·a)·a^T
=E-4aa^T+4aa^T
=E


So, H is an orthogonal matrix.

Let u be an n-dimensional unit column vector. Let H=E-2u^t be an orthogonal matrix.

Because u is a unit vector,
U^t*u=1H (T) H=(E-2uu^t) T*(E-2uu^t)=(E-2uu^t)*(E-2uu^t)=E-4uu^t+4uu^t*uu^t=E-4uu^t+4uu^t=E
Please ask if you don't understand. Please choose the best answer. Thank you.

Since u is a unit vector,
U^t*u=1H (T) H=(E-2uu^t) T*(E-2uu^t)=(E-2uu^t)*(E-2uu^t)=E-4uu^t+4uu^t*uu^t=E-4uu^t+4uu^t=E
Please ask if you don't understand. Please choose the best answer. Thank you.

Let x be n-dimensional column vector, x^t*x=1, let a=e-2x*x^t, and prove that a is an orthogonal matrix.

Use orthogonal matrix definition to verify. The economic mathematics team will help you to solve. Please evaluate in time.

Given the point A (-1,2) B (1,2) C (-2,1) D (3,4), the projection of vector AB on cd is

AB=(2,1), CD=(5,5)
AB·CD=15
|AB |=√5,|CD |=5√2
The projection of vector AB in the direction of vector CD is
AB·CD/|CD|=3√2/2