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True or False?

A matrix with linearly independent row vectors and linearly independent column vectors is a square.

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1 Answer 1

up vote 3 down vote accepted

True.

Suppose a matrix of $m\times n$ size has $m$ linearly independent row vectors and $n$ linearly independent column vectors. We know that the row space and the column space must have the same dimension.

Because of linear independence, $dim(R) = m, dim(C) = n$, $R, C$ stand for row and column space. And $$dim(R) = dim(C)$$Thus $$m = n$$

To see why the row space and the column space have the same dimension, consider the rank of a matrix.

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thanks for your help, but your explanation is rather useless. Staying dim (R) = dim (c) doesn't explain why the matrix is a square, it is just another way of saying the matrix is n x n –  Johnathon Svenkat Feb 14 '13 at 17:24
    
$dim(R)=dim(C)$ is a property satisfied by all matrices, and it is because of that property such that the matrices described in your question are squares. I am simply laying out the logic. –  Enzo Feb 14 '13 at 17:26
    
To see why this property is true, consider the last sentence in my answer. –  Enzo Feb 14 '13 at 17:27
    
ok great. thanks –  Johnathon Svenkat Feb 14 '13 at 18:41

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