Question

The function f₁ and f₂ are linearly independent if there is a real number X such that k₁f₁(x) + k₂f₂(x)=0

(2) Prove that if V is a subspace of a vector space W and if V is infinite demensial then so in W.

(3) Prove that if A is digonalizble matrix then the rank of A is the number of non-zero eigen values of A.

Answer 1

2) By Theorem:

A vector space $V$ is infinite dimensional if and only if there is a sequence of vectors $v_1,v_2,v_3,v_4,...$ such that for all $n\isin Z^+$ the subsequence $v_1,v_2,...,v_n$

is linearly independent.

Since V is a subspace of a vector space W, then the given sequence of vectors exists in W also. So, W is infinite dimensional.

3) Let's say D is the diagonalized form of A. Then the diagonal elements of D are precisely the eigenvalues of A. The rank of D is the number of linearly independent columns. Obviously this equals the number of non-zero eigenvalues. Since the rank of A and the rank of D are the same, the conclusion follows.