Difference between Rank of a matrix and Rank of a Linear transformation

Hello all,
I have a doubt in calculating the rank of a matrix and rank of a linear transformation
In the first picture the instructor has told that the rank of a matrix:

rank = 2 - (dimension of solution space)
and someone also explain what a dimensional space is ?

but in the the second picture the instructor has told that rank is:
rank = No Of Dimensions

Hi @sai_shankar1!

The solution space is also known as the kernel of the Linear Transformation. In other words it is the set of points x, such that T(x) = 0. Where T is the linear transformation.

When looking the matrices as linear transformations, the rank is the dimension that it is generated by the linear transformation. So if M is a matrix, we can look at it as a linear transformation, so the set of points M \cdot x is a space and its dimension is the rank of the linear transformation.

Was that clear?


Thank you so much for explaining @lucas.coutinho