Answer: true
Explanation:
For an m*n matrix, the column space of A will be a space formed by the lineal combination of all the columns of A.
column space = a1*c1 + a2*c2 + ...
where a1, a2, ... are scalars, and c1 is the vector of column 1.
Then we should write:
Column space = a1*(A₁₁, A₂₁, A₃₁, ...) + a2*(A₁₂, A₂₂, A₃₂, ...) + ...
Now, the transpose is defined as:
[At]₁₃ = A₃₁
Here i used the element with subindex 3 and 1, but is the same for every subindex.
Notice that if A is m*n, then [At] is n*m
Now, the row space of [At] will be, same as before.
Row space = b1*r1 + b2*r2 + ...
Where b1, b2, ... are scalars and the r's are the vector of each row.
= b1*( [At]₁₁ , [At]₁₂, [At]₁₃, ...) + b2*([At]₂₁, [At]₂₂, [At]₂₃, ...) + ...
Now we replace each term of the transpose by the associated element in the original matrix.
= b1*( A₁₁, A₂₁, A₃₁, ...) + b2*(A₂₁, A₂₂, ...) + ....
If we take:
b1 = a1, b2 = a2, b3 = a3, ...
We will have that the row space of [At] is the same as the column space of A.