To find the dimension of the set $H$ and determine a basis for it, we need to consider the matrix formed by the given columns and perform row reduction to identify the pivot columns. The number of pivot columns corresponds to the dimension of $H$, and the columns corresponding to these pivots form a basis for $H$.

Given:

\begin{pmatrix} -2 \\ 1 \\ 1 \\ -3 \end{pmatrix}, \begin{pmatrix} -9 \\ 4 \\ 3 \\ -9 \end{pmatrix}, \begin{pmatrix} -33 \\ 15 \\ 12 \\ -36 \end{pmatrix}, \begin{pmatrix} -22 \\ 10 \\ 8 \\ -24 \end{pmatrix} \right\}$$ Let's form the matrix with these columns and perform row reduction. Matrix $$A$$ is: $$A = \begin{pmatrix} -2 & -9 & -33 & -22 \\ 1 & 4 & 15 & 10 \\ 1 & 3 & 12 & 8 \\ -3 & -9 & -36 & -24 \end{pmatrix}$$ We will now perform row reduction to find the pivot columns and the basis.The matrix $$H$$ in row reduced echelon form (RREF) is: $$\begin{pmatrix} 1 & 0 & 3 & 2 \\ 0 & 1 & 3 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$ ### Dimension of $$H$$ The dimension of $$H$$ is equal to the number of pivot columns in the RREF, which is **2**. ### Basis for $$H$$ The columns corresponding to the pivot columns in the original matrix form a basis for $$H$$. Therefore, the basis for $$H$$ is: $$\left\{ \begin{pmatrix} -2 \\ 1 \\ 1 \\ -3 \end{pmatrix}, \begin{pmatrix} -9 \\ 4 \\ 3 \\ -9 \end{pmatrix} \right\}$$ This set of vectors forms a basis for the column space of $$H$$. Do you have any questions or would you like more details on the process? Here are some related questions to explore: 1. What is the significance of finding the dimension of a vector space? 2. How can you use the basis of a space to express any vector in that space? 3. How does the row reduced echelon form help in determining linear independence? 4. What is the relationship between the rank of a matrix and the dimension of its column space? 5. Can the dimension of $$H$$ change if the columns are rearranged? 6. How do elementary row operations affect the basis of a matrix? 7. What are the implications of having zero rows in the RREF of a matrix? 8. How does one verify that the vectors in the basis are linearly independent? **Tip:** When determining a basis from a matrix, always check that the pivot columns correspond to linearly independent vectors from the original set.