Question

In: Advanced Math

Write up a full proof of the fact that every k-dimensional subspace of R^n is the...

Write up a full proof of the fact that every k-dimensional subspace of R^n is the intersection of (n-k) hyperplanes. Tip: If you don't know how to start, begin by summarizing your answers to the previous problems on this lab.

Solutions

Expert Solution

Colby Messinger answered 11 months ago
Next > < Previous

Related Solutions

(3) (a) Show that every two-dimensional subspace of R3 is the kernel of some linear transformation...
(3) (a) Show that every two-dimensional subspace of R3 is the kernel of some linear transformation T : R3 → R. [Hint: there are many possible ways to approach this problem. One is to use the following fact, typically introduced in multivariable calculus: for every plane P in R3, there are real numbers a, b, c, d such that a point (x,y,z) belongs to P if and only if it satisfies the equation ax+by+cz = d. You may use this...
Let S be the two dimensional subspace of R^4 spanned by x = (1,0,2,1) and y...
Let S be the two dimensional subspace of R^4 spanned by x = (1,0,2,1) and y = (0,1,- 2,0) Find a basis for S^⊥
4. Here is a fact about permutations: (**) nPk = n!/(n-k)!, for all k € ≤...
4. Here is a fact about permutations: (**) nPk = n!/(n-k)!, for all k € ≤ n. Let’s prove this via mathematical induction for the fixed case k=3. (i) Write clearly the statement (**) we wish to prove. Be sure your statement includes the phrase “for all n” . (ii) State explicitly the assumption in (**) we will thus automatically make about k=2. (iii) Now recall that to prove by induction means to show that If mPk = m!/(m-k)! is...
The n- dimensional space is colored with n colors such that every point in the space...
The n- dimensional space is colored with n colors such that every point in the space is assigned a color. Show that there exist two points of the same color exactly a mile away from each other.
Write a combinatorial proof for 1 n + 2 ( n − 1 ) + 3...
Write a combinatorial proof for 1 n + 2 ( n − 1 ) + 3 ( n − 2 ) + ⋯ + ( n − 1 ) 2 + n 1 = ( n + 2 choose 3 ) .
Let W be a subspace of R^n, and P the orthogonal projection onto W. Then Ker...
Let W be a subspace of R^n, and P the orthogonal projection onto W. Then Ker P is W^perp.
How to proof: Matrix A have a size of m×n, and the rank is r. How...
How to proof: Matrix A have a size of m×n, and the rank is r. How can we rigorous proof that the dimension of column space are always equal to the dimensional of row space? (I can use many examples to show this work, but how to proof rigorously?)
Write up a formal proof that the angle bisectors of a triangle are concurrent, and that...
Write up a formal proof that the angle bisectors of a triangle are concurrent, and that the point of concurrency (the incenter) is equidistant from all three sides.
18. If rS = .25, n = 25, set up an inferential proof with the 7...
18. If rS = .25, n = 25, set up an inferential proof with the 7 steps to test the significance of the correlation. (4 points)
In the proof of bolzano-weierstrass theorem in R^n on page 56 of "Mathematical Analysis" by Apostol,...
In the proof of bolzano-weierstrass theorem in R^n on page 56 of "Mathematical Analysis" by Apostol, should the inequality be a/2^(m-2) < r/sqrt(n) or something related to n? a/2^(m-2) < r/2 seems not enough
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT