We have linearly independent vectors e_1,e_2,...,e_(m+1) in RR^n. I would like to prove that among their linear combinations, there is a nonzero vector whose first m coordinates are zero. From independence we can get that m+1 <= n. Maybe we can build some system, but what next?

trkalo84 2022-09-15 Answered
We have linearly independent vectors e 1 , e 2 , . . . , e m + 1 R n . I would like to prove that among their linear combinations, there is a nonzero vector whose first m coordinates are zero.
From independence we can get that m + 1 n. Maybe we can build some system, but what next?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Simon Zhang
Answered 2022-09-16 Author has 7 answers
A linear combination of e 1 , e 2 , . . . e m is of the form: a 1 e 1 + a 2 e 2 + . . . a m + 1 e m + 1 . If you write each ei out, you see that the condition where the first m coordinates are zero is equivalent to solving an equation of m+1 variables over m equations, which you can always find a solution for.
Another way of saying this is span { e 1 , e 2 . . . e m + 1 } is of dimension m+1, while the vector space of vectors with the first m coordinates zero has dimension n−m. If their intersection consisted only of the 0 vector, dimension of whole space is n−m+m+1, contradiction.

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-05-29
Which of the following expressions are meaningful? Which are meaningless? Explain.
a) (ab)c
(ab)c has ? because it is the dot product of ?.
b) (ab)c
(ab)c has ? because it is a scalar multiple of ?.
c) |a|(bc)
|a|(bc) has ? because it is the product of ?.
d) a(b+c)
a(b+c) has ? because it is the dot product of ?.
e) ab+c
ab+c has ? because it is the sum of ?.
f) |a|(b+c)
|a|(b+c) has ? because it is the dot product of ?.
asked 2021-05-17
Find the scalar and vector projections of b onto a.
a=(4,7,4),b=(3,1,1)
asked 2021-05-29
Find a vector equation and parametric equations for the line segment that joins P to Q.
P(0, - 1, 1), Q(1/2, 1/3, 1/4)
asked 2021-02-11
Let F be a fixed 3x2 matrix, and let H be the set of all matrices A in M2×4 with the property that FA = 0 (the zero matrix in M3×4). Determine if H is a subspace of M2×4
asked 2022-08-18
Three vectors, V 1 , V 2 , V 3 are in the R 2 plane where V 1 + V 2 + V 3 = 0 and the magnitudes of these vectors are the same. Show that the angle between any two of these vectors is 120 degrees.
My try:
V 1 + V 2 + V 3 = 0 , so V 1 + V 2 = V 3 . This would mean that ( V 1 + V 2 ) 2 = V 3 2 which is | V 1 | + | V 2 | + 2 V 1 V 2 = | V 3 | . But since the magnitude is the same, we get 2 | V 1 | + 2 V 1 V 2 = | V 1 | which is 2 V 1 V 2 = | V 1 |
The formula for the angle is cos ( a ) = V 1 V 2 | V 1 | | V 2 | = 0.5 | V 1 | | V 1 | 2 = 0.5 | V 1 |
I know that the solution would be a triangle and that cos ( a ) = 0.5
But that would mean that my angle is dependent on V 1 which isn't the case. So what am I doing wrong?
asked 2022-07-24
Two lines are given by the equation x = 10 - 8t, y=1+8t, z=15 +2t and x=6-8t, y=2+2t, z=3+6t What is the shortest distance between these two lines?
asked 2022-08-22
I have a line in the vector form: ( x , y , z ) = t ( 1 , 1 , 1 ) + ( 0 , 0 , 1 ) and I need to find a vector that represents all the points whose distance from this line equals 1. I started calculating the distance between a point ( a , b , c ) and this line using the vetorial product... and I got a big equation in terms of a, b and c. I think this equation describes a cylinder whose centerline is ( x , y , z ) = t ( 1 , 1 , 1 ) + ( 0 , 0 , 1 ). But I don't know how to get rid of this equation and obtain the vector...
Can anyone help me?

New questions