- #1
lalligagger
- 14
- 0
Sorry all my vectors look like superscripts, don't know what that's about.
What is wrong with the following argument?
Let A be an arbitrary m x n matrix. The vector A[tex]\vec{x}[/tex] is obviously in CA so it can't be in N(AT) unless it's the zero vector, since CA is orthogonal to N(AT). Thus the only solution to ATA[tex]\vec{x}[/tex]=[tex]\vec{0}[/tex] is [tex]\vec{x}[/tex]=[tex]\vec{0}[/tex] and ATA is an invertible matrix (by FTIM).
The Fundamental Theorem of Invertible Matrices.
I don't know if I understand the (incorrect) reasoning behind this argument. Mainly, I don't understand the connection between the end of the second sentence and the first half of the last sentence. What does CA and N (AT) being orthogonal have to do with the equation ATA[tex]\vec{x}[/tex]=[tex]\vec{0}[/tex]?
Homework Statement
What is wrong with the following argument?
Let A be an arbitrary m x n matrix. The vector A[tex]\vec{x}[/tex] is obviously in CA so it can't be in N(AT) unless it's the zero vector, since CA is orthogonal to N(AT). Thus the only solution to ATA[tex]\vec{x}[/tex]=[tex]\vec{0}[/tex] is [tex]\vec{x}[/tex]=[tex]\vec{0}[/tex] and ATA is an invertible matrix (by FTIM).
Homework Equations
The Fundamental Theorem of Invertible Matrices.
The Attempt at a Solution
I don't know if I understand the (incorrect) reasoning behind this argument. Mainly, I don't understand the connection between the end of the second sentence and the first half of the last sentence. What does CA and N (AT) being orthogonal have to do with the equation ATA[tex]\vec{x}[/tex]=[tex]\vec{0}[/tex]?