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Problem 3 Solutions

Given matrix $A = \begin{bmatrix} 1 & 1 & 0 & 1 & 0 \\ 0 & 4 & 0 & 12 & 4 \\ -1 & 0 & 0 & 3 & 1 \\ 0 & 2 & 0 & 4 & 2 \end{bmatrix}$, we have the following tasks:

(a) Find a basis for Col A (Column Space of A)

To find a basis for the column space of $A$, we must determine the columns of $A$ that are linearly independent. We do this by row reducing $A$ to its echelon form and identifying the pivot columns. The columns in $A$ corresponding to the pivots will form a basis for the column space.

1. Perform row reduction on $A$ to reach an echelon form.
2. Identify the pivot columns and use them as a basis.

(b) Find a basis for Nul A (Null Space of A)

To find a basis for the null space of $A$, solve $A\mathbf{x} = 0$:

1. Set up the system $A\mathbf{x} = 0$ and row-reduce $A$ to find the solutions.
2. The free variables will determine the basis vectors for the null space.

(c) Find a basis for Row A (Row Space of A)

The row space of $A$ is spanned by the linearly independent rows of $A$.

1. Row-reduce $A$ to echelon form.
2. The non-zero rows in the echelon form of $A$ will form a basis for the row space.

(d) Find a different basis for Col A that includes $\begin{bmatrix} 2 \\ 4 \\ -1 \\ 2 \end{bmatrix}$ as one of the basis vectors.

To incorporate $\begin{bmatrix} 2 \\ 4 \\ -1 \\ 2 \end{bmatrix}$ into the basis for Col A:

1. Verify if $\begin{bmatrix} 2 \\ 4 \\ -1 \\ 2 \end{bmatrix}$ is in Col A by expressing it as a linear combination of the columns of $A$.
2. Construct a new basis by including $\begin{bmatrix} 2 \\ 4 \\ -1 \\ 2 \end{bmatrix}$ and linearly independent vectors from the original basis.

(e) Show that $\begin{bmatrix} 3 \\ 5 \\ 0 \\ -5 \\ 2 \end{bmatrix}$ is in Row A.

To verify if $\begin{bmatrix} 3 \\ 5 \\ 0 \\ -5 \\ 2 \end{bmatrix}$ is in Row A:

1. Check if it can be expressed as a linear combination of the rows of $A$.
2. Alternatively, transpose $A$ and confirm if $\begin{bmatrix} 3 \\ 5 \\ 0 \\ -5 \\ 2 \end{bmatrix}$ is in the column space of $A^T$.

If you need detailed calculations or further breakdown for each part, feel free to ask!

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Follow-up Questions:

1. How do you determine linear independence among vectors?
2. What is the significance of finding a basis for the column space and null space of a matrix?
3. How does the rank of a matrix relate to its column space and null space?
4. Can a vector belong to both the column space and null space of a matrix?
5. What methods exist to determine if a vector lies in a particular vector space?

Tip: Remember that the row space, column space, and null space are all fundamental subspaces related to a matrix and provide insights into its structure and solution properties.