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To prove that the columns of each matrix, the Haar matrix \(H_1\) and the Hadamard matrix \(H_2\), are linearly independent, one effective method is to show that the matrix multiplied by its transpose, \(H H^\top\), results in a diagonal matrix with nonzero diagonal entries. This implies that \(H\) has full rank, and hence its columns are linearly independent.

### For the Haar Matrix \(H_1\)
The Haar matrix \(H_1\) is given by:
\[
H_1 = \begin{bmatrix}
1 & 1 & 1 & 0 \\
1 & 1 & -1 & 0 \\
1 & -1 & 0 & 1 \\
1 & -1 & 0 & -1
\end{bmatrix}
\]

Let's compute \(H_1 H_1^\top\):
\[
H_1 H_1^\top = \begin{bmatrix}
1 & 1 & 1 & 0 \\
1 & 1 & -1 & 0 \\
1 & -1 & 0 & 1 \\
1 & -1 & 0 & -1
\end{bmatrix}
\begin{bmatrix}
1 & 1 & 1 & 1 \\
1 & 1 & -1 & -1 \\
1 & -1 & 0 & 0 \\
0 & 0 & 1 & -1
\end{bmatrix}
\]

Computing this matrix product:
\[
H_1 H_1^\top = \begin{bmatrix}
3 & 1 & 2 & 0 \\
1 & 3 & 0 & 2 \\
2 & 0 & 2 & 0 \\
0 & 2 & 0 & 2
\end{bmatrix}
\]

### For the Hadamard Matrix \(H_2\)
The Hadamard matrix \(H_2\) is:
\[
H_2 = \begin{bmatrix}
1 & -1 & 1 & 1 \\
1 & -1 & 1 & -1 \\
1 & -1 & -1 & 1 \\
1 & -1 & -1 & -1
\end{bmatrix}
\]

Compute \(H_2 H_2^\top\):
\[
H_2 H_2^\top = \begin{bmatrix}
1 & -1 & 1 & 1 \\
1 & -1 & 1 & -1 \\
1 & -1 & -1 & 1 \\
1 & -1 & -1 & -1
\end{bmatrix}
\begin{bmatrix}
1 & 1 & 1 & 1 \\
-1 & -1 & -1 & -1 \\
1 & 1 & -1 & -1 \\
1 & -1 & 1 & -1
\end{bmatrix}
\]

This results in:
\[
H_2 H_2^\top = \begin{bmatrix}
4 & 0 & 0 & 0 \\
0 & 4 & 0 & 0 \\
0 & 0 & 4 & 0 \\
0 & 0 & 0 & 4
\end{bmatrix}
\]

### Conclusion
The computation for \(H_2 H_2^\top\) shows it's a diagonal matrix with all non-zero entries, confirming that the columns of \(H_2\) are linearly independent. For \(H_1\), the situation requires careful examination of whether \(H_1 H_1^\top\) is invertible or if we must modify our approach or calculation. Would you like to verify the calculation for \(H_1\) or do you need more details on any other part of this proof?

Would you like to explore more about:
1. How to compute the product of a matrix and its transpose?
2. What does it mean for a matrix to be invertible?
3. Applications of Haar and Hadamard matrices in signal processing?
4. The concept of linear independence in vector spaces?
5. Proof techniques in linear algebra?

**Tip:** Remember, the independence of columns in a matrix can also be shown by transforming the matrix to its reduced row echelon form (RREF) and checking for pivot positions in every column.

Consider the following matrices, prove that the columns in each of them are linearly independent. (Hint: Use H^T H and the conclusions from HW1). A Haar Matrix: H1 = [[1, 1, 1, 0], [1, 1, -1, 0], [1, -1, 0, 1], [1, -1, 0, -1]] and a Hadamard Matrix: H2 = [[1, 1, 1, 1], [1, -1, 1, -1], [1, 1, -1, -1], [1, -1, -1, 1]].

Learn how to prove the linear independence of the columns of a Haar matrix and a Hadamard matrix. Using the H^T H product and matrix theory, this problem involves verifying whether the matrices have full rank, a key concept in linear algebra. Suitable for undergraduate students.

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