Dimension of Solution Space of Homogeneous System

I'm quite new to Linear Algebra.So, I hope someone would help me with this.

For linear systems with n unknowns and with matrix of coefficients A, if the rank of A is r then following holds

The Vector Space of Solutions of the associated Homogeneous System has Dimension n − r

There is already question about it here

Why is dimension of solution space of homogeneous equations n-r? but it doesn't provide proof to it.

Could someone point to proof of this result (Hints would be more appreciated )?

1 Answer

This follows from the fact that if $\ \mathbf{T}: V \to W$ is a linear transformation from finite dimensional vector spaces $V$ and $W$, then we can represent $\mathbf{T}$, in terms of the bases of $V$ and $W$, by a matrix and the dimension formula: $$\mathsf{dim} V = \mathsf{dim}(ker \ \mathbf{T}) \ + \ \mathsf{dim}( Img \ \mathbf{T})$$

Sketch of the proof: Assume $\ \mathsf{dim}V = n \$. If $\ ( \alpha_1,\dots, \alpha_k)$ is a basis for the subspace $\ ker \mathbf{T}$, then this basis can be extended to a basis of $V$, say $( \alpha_1,\dots, \alpha_k, \beta_1, \dots, \beta_{n-k})$. If we let $$w_i = \mathbf{T}(\beta_i) \quad for \ i = 1, \dots, n-k$$ and show that the set $\{w_1, \dots, w_{n-k}\}$ is linearly independent and spans the image of the transformation, then the dimension of the image is $n-k$, proving the dimension formula.