Math Problem Statement
FYE20-grammiki-B.pdf
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solve 3
Solution
The document contains a mathematical exam in Greek focused on Linear Algebra. Let's solve problem 3, which involves finding the dimension and a basis for the row space and column space of a given matrix .
Given matrix :
1 & 2 & 0 & 2 & 1 \\ -1 & -2 & 1 & 1 & 0 \\ 1 & 2 & -3 & -7 & -2 \end{pmatrix}$$ ### Row Space To find the row space, we need to row reduce the matrix $$A$$ to its row echelon form. First, we start with the given matrix: $$\begin{pmatrix} 1 & 2 & 0 & 2 & 1 \\ -1 & -2 & 1 & 1 & 0 \\ 1 & 2 & -3 & -7 & -2 \end{pmatrix}$$ Perform row operations to simplify it: 1. Add the first row to the second row: $$R2 = R2 + R1$$ $$\begin{pmatrix} 1 & 2 & 0 & 2 & 1 \\ 0 & 0 & 1 & 3 & 1 \\ 1 & 2 & -3 & -7 & -2 \end{pmatrix}$$ 2. Subtract the first row from the third row: $$R3 = R3 - R1$$ $$\begin{pmatrix} 1 & 2 & 0 & 2 & 1 \\ 0 & 0 & 1 & 3 & 1 \\ 0 & 0 & -3 & -9 & -3 \end{pmatrix}$$ 3. Add 3 times the second row to the third row: $$R3 = R3 + 3R2$$ $$\begin{pmatrix} 1 & 2 & 0 & 2 & 1 \\ 0 & 0 & 1 & 3 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}$$ The row echelon form shows that the first and second rows are linearly independent, and the third row is zero. ### Basis and Dimension - **Row space** basis: The non-zero rows of the row echelon form matrix form a basis for the row space. $$\left\{ \begin{pmatrix} 1 & 2 & 0 & 2 & 1 \end{pmatrix}, \begin{pmatrix} 0 & 0 & 1 & 3 & 1 \end{pmatrix} \right\}$$ - **Dimension of the row space**: The number of non-zero rows in the row echelon form, which is 2. ### Column Space The columns of the original matrix corresponding to the leading ones in the row echelon form form a basis for the column space. $$\left\{ \begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}, \begin{pmatrix} 2 \\ -2 \\ 2 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ -3 \end{pmatrix} \right\}$$ - **Dimension of the column space**: The number of leading ones in the row echelon form, which is 3. **Summary:** - Dimension of the row space: 2 - Basis for the row space: $$\{(1, 2, 0, 2, 1), (0, 0, 1, 3, 1)\}$$ - Dimension of the column space: 3 - Basis for the column space: $$\{(1, -1, 1), (2, -2, 2), (0, 1, -3)\}$$ Would you like more details or have any questions? Here are 5 numbered questions for further exploration: 1. How do you determine the rank of a matrix? 2. What is the difference between row space and column space? 3. How do you find the null space of a matrix? 4. What is the significance of the row echelon form? 5. Can you explain how to find the eigenvalues of a matrix? **Tip:** Always double-check your row operations during matrix reduction to avoid errors in finding the basis and dimension.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Row Space
Column Space
Basis
Formulas
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Theorems
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Suitable Grade Level
Advanced College Level