Linked Questions

15
votes
3answers
8k views

What's the meaning of the transpose? [duplicate]

I don't understand the motivation of the transpose (or better yet, I haven't even seen one). It feels like just something pulled out of a hat. Thinking about it makes it seem like a product of being ...
11
votes
3answers
3k views

What's the intuition of the transpose of a matrix? [duplicate]

I know the transpose is to swap the columns and rows of a matrix. And $A^T$$A$ is a symmetric matrix which elements are the inner product of each column of $A$. But I didn't understand the intuition ...
2
votes
0answers
58 views

Geometrical interpretation of transpose of a matrix [duplicate]

How can we interpret the transpose of a matrix, that represents the cordinates of a vector with respect to a basis, geometrically?
56
votes
5answers
66k views

Determinant of transpose?

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns could someone give a geometric interpretation of the property? Thanks!
8
votes
1answer
3k views

Truly intuitive geometric interpretation for the transpose of a square matrix

I'm looking for an easily understandable interpretation for a transpose of a square matrix A. An intuitive visual demonstration, how $A^{T}$ relates to A. I want to be able to instantly visualize in ...
3
votes
2answers
275 views

Why is column space on the vertical in a matrix?

Why is the "column space" on the vertical in a matrix? In my mind the column space is that space that the vectors in the matrix have created. I mean, for example take the equations: ...
2
votes
1answer
861 views

Geometric interpretation of matrices

I'm interested in knowing some geometric interpretation of matrices. Can you suggest any lecture note or textbook or anything else about it? I've just finished an undergraduate course in linear ...
1
vote
1answer
360 views

graphical interpretation of transpose?

I am working on a problem and I think that knowing if there is some sort of pattern to a transpose graphically to its normal matrix could potentially be useful. Is there any reflection, rotation, etc ...
1
vote
1answer
182 views

Geometric interpretation of the adjoint - what does it mean that the graphs are orthogonal

I am trying to understand the adjoint of a linear operator geometrically. Since the graph of the adjoint can be constructed as the orthogonal complement of a "rotated" copy of the graph of the ...
0
votes
0answers
68 views

Adjoints : the mysterious linear operator!

The definition of Adjoints says "Let $T$be a linear operator on an inner product space $V$. Then we say that $T$ has an adjoint on $V$ if there exists a linear operator $T^*$ on $V$ such that $\langle ...
1
vote
0answers
44 views

Transpose of matrices

I have been trying to find an intuition for the transpose of a matrix. I have also read the posts in mathstackexhange but unfortunately it couldn’t help me.Can anyone please explain what is the idea ...
0
votes
0answers
29 views

What is the geometric meaning of matrix transpose? [duplicate]

I had to prove that det$A$ = det$A^T$. What is the geometry behind that?