I have the following question in an exercise of linear fitting.
We have [MATH]n[/MATH] linearly independent known vectors [MATH]\textbf{a}_i[/MATH], [MATH]i = 1, . . . , n[/MATH], a known vector [MATH]\textbf{b}[/MATH], and an unknown vector [MATH]\textbf{x}[/MATH]. All vectors are of dimension [MATH]m[/MATH]. Under what conditions is there a solution to the following equation? Express in terms of [MATH]m[/MATH] and [MATH]n[/MATH].
[MATH]\sum_{i=1}^{n} \textbf{a}_i\textbf{a}_i^T \textbf{x}= \textbf{b}[/MATH]
My attempt:
Let [MATH]\textbf{A} = [\textbf{a}_1, ..., \textbf{a}_n][/MATH], [MATH]m × n[/MATH] matrix
So, [MATH]\textbf{A}\textbf{A}^T \textbf{x}= \sum_{i=1}^{n} \textbf{a}_i\textbf{a}_i^T \textbf{x} = \textbf{b}[/MATH]If [MATH]m = n[/MATH], [MATH]\textbf{A}\textbf{A}^T[/MATH] is full rank and has a unique solution.
Will it have a solution if [MATH]m<n[/MATH] or [MATH]m>n[/MATH]? Why or why not? What other conditions am I missing? Am I missing condition for vector b to lie in column space of matrix [MATH]\textbf{A}\textbf{A}^T[/MATH] ?
We have [MATH]n[/MATH] linearly independent known vectors [MATH]\textbf{a}_i[/MATH], [MATH]i = 1, . . . , n[/MATH], a known vector [MATH]\textbf{b}[/MATH], and an unknown vector [MATH]\textbf{x}[/MATH]. All vectors are of dimension [MATH]m[/MATH]. Under what conditions is there a solution to the following equation? Express in terms of [MATH]m[/MATH] and [MATH]n[/MATH].
[MATH]\sum_{i=1}^{n} \textbf{a}_i\textbf{a}_i^T \textbf{x}= \textbf{b}[/MATH]
My attempt:
Let [MATH]\textbf{A} = [\textbf{a}_1, ..., \textbf{a}_n][/MATH], [MATH]m × n[/MATH] matrix
So, [MATH]\textbf{A}\textbf{A}^T \textbf{x}= \sum_{i=1}^{n} \textbf{a}_i\textbf{a}_i^T \textbf{x} = \textbf{b}[/MATH]If [MATH]m = n[/MATH], [MATH]\textbf{A}\textbf{A}^T[/MATH] is full rank and has a unique solution.
Will it have a solution if [MATH]m<n[/MATH] or [MATH]m>n[/MATH]? Why or why not? What other conditions am I missing? Am I missing condition for vector b to lie in column space of matrix [MATH]\textbf{A}\textbf{A}^T[/MATH] ?