Math Problem Statement
e are given the matrix 𝐴 A:
𝐴
[ 2 3 1 4 1 1 1 1 1 1 1 2 ] A=
2 1 1
3 1 1
1 1 1
4 1 2
Find a basis for the image imA
Solution
To find a basis for the image of a matrix , we determine the set of linearly independent columns in . Here's the step-by-step solution:
Step 1: Write the Matrix
\begin{bmatrix} 2 & 3 & 1 & 4 \\ 1 & 1 & 1 & 2 \\ 1 & 1 & 1 & 2 \end{bmatrix}$$ --- ### Step 2: Perform Column Reduction to Identify Independent Columns We'll reduce $$A$$ to row-echelon form to determine which columns are linearly independent. 1. Start with the matrix $$A$$: $$\begin{bmatrix} 2 & 3 & 1 & 4 \\ 1 & 1 & 1 & 2 \\ 1 & 1 & 1 & 2 \end{bmatrix}$$ 2. Subtract $$\frac{1}{2}$$ of the first row from the second and third rows to eliminate the first element in those rows: $$R_2 \to R_2 - \frac{1}{2}R_1, \quad R_3 \to R_3 - \frac{1}{2}R_1$$ Result: $$\begin{bmatrix} 2 & 3 & 1 & 4 \\ 0 & -0.5 & 0.5 & 0 \\ 0 & -0.5 & 0.5 & 0 \end{bmatrix}$$ 3. Subtract $$R_2$$ from $$R_3$$ to eliminate the second row: $$R_3 \to R_3 - R_2$$ Result: $$\begin{bmatrix} 2 & 3 & 1 & 4 \\ 0 & -0.5 & 0.5 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ --- ### Step 3: Identify Independent Columns In row-echelon form, we see that: - Column 1 and column 2 correspond to pivot columns. Thus, the first two columns of the original matrix $$A$$ are linearly independent and form a basis for the image. --- ### Step 4: Basis for $$\text{im} A$$ The basis for the image of $$A$$ is given by: $$\text{Basis} = \left\{ \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix} \right\}.$$ --- Would you like further explanation on any of these steps? ### Follow-up Questions: 1. How do we determine if a column is linearly dependent? 2. What is the connection between the rank of a matrix and its image? 3. How can the concept of null space help in understanding the image? 4. Can the row reduction process change the basis vectors? 5. What is the dimension of the image, and why does it matter? ### Tip: Always verify the independence of columns in your basis by checking if their determinant is non-zero (when possible).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrices
Column Space
Row Echelon Form
Linear Independence
Formulas
Row Echelon Form
Basis of a Subspace
Theorems
Rank-Nullity Theorem
Column Space Theorem
Suitable Grade Level
Grades 10-12