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### Linearly Independent columns and invertibility of transpose [duplicate]

If a matrix $A$ has linearly independent columns, why is $A^TA$ invertible? I think it has to do with $A^TA$ being square and that $A^TAx$=0 only has the trivial solution, but I don't really know ...
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### $A\in \mathbb{R}^{m\times n}, m\ge n$ and $\operatorname{rank}(A) = n$. Prove that $A^tA$ is nonsingular

Let $A\in \mathbb{R}^{m\times n}, m\ge n$ and $\operatorname{rank}(A) = n$. Prove that $A^tA$ is nonsingular I found this answer https://math.stackexchange.com/a/1840814/166180 which gives some ...
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### Proving change of basis matrix from $\mathcal B$ to $\mathcal B'$ is $\mathbf P$=$(\mathbf Y^T \mathbf Y)^{-1}\mathbf Y^T\mathbf X$

If $\mathcal B=${${x_1,x_2,...,x_n}$} and $\mathcal B'=${${y_1,y_2,...,y_n}$} are bases for an $n$-dimensional subspace of $\mathcal V \subset R^m$. Say I let $X_{mxn}$ and $Y_{mxn}$ be the matrices ...
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### Show that if $A \in \mathbb{C}^{m \times n}$ is of full rank, then null$(A^*)$ is the orthogonal complement of range$(A)$.

Show that if $A \in \mathbb{C}^{m \times n}$ is of full rank, then null$(A^*)$ is the orthogonal complement of range$(A)$. I saw the above fact listed in this linked MSE proof. My attempt to ...
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### Show $X^TX$ is not invertible [duplicate]

Suppose that the first column of $X \in R^{N \times (p+1)}$ is full of $1$s. Show that in the following cases, there exists no inverse matrices for $X^TX$. (a). $N < p+1$ (b). $N \ge p+1$ and two ...
If $X$ is a $N \times D$ matrix with $(D\gg N)$ with $\operatorname{rank}(X) = N$, what is $\operatorname{rank}(X^T \cdot X)$ where $X^T$ is the transpose matrix of $X$? I am little new to linear ...
### Why is $\sum_i^n \mathbf{x_i}\mathbf{x_i}'$ positive definite?
Let $\mathbf{x_i}=[ x_{i1}, x_{i2}, x_{i3} ]'$. Why can we say that $\sum_i^n \mathbf{x_i}\mathbf{x_i}'$ is positive definite?