Math Problem Statement

Trong R^3 cho các vectơ u1 = (1,2,1), u2 = (-1,1,2), u3 = (2,1,-1). Tìm một cơ sở và số chiều của không gian con W = <{u1, u2, u3}>.

Solution

The problem is asking us to find a basis and the dimension of the subspace WW spanned by the vectors u1=(1,2,1)\mathbf{u}_1 = (1, 2, 1), u2=(−1,1,2)\mathbf{u}_2 = (-1, 1, 2), and u3=(2,1,−1)\mathbf{u}_3 = (2, 1, -1) in R3\mathbb{R}^3. Let’s solve this step by step:

Step 1: Write the vectors as rows of a matrix

To determine the dimension and basis of WW, we will check the linear independence of the vectors by forming a matrix where each row is one of the given vectors:

1 & 2 & 1 \\ -1 & 1 & 2 \\ 2 & 1 & -1 \end{bmatrix}.$$ ### Step 2: Row reduce the matrix We row reduce $$A$$ to row echelon form to determine the rank, which corresponds to the dimension of $$W$$. 1. The original matrix: $$\begin{bmatrix} 1 & 2 & 1 \\ -1 & 1 & 2 \\ 2 & 1 & -1 \end{bmatrix}.$$ 2. Add $$R_1$$ to $$R_2$$ (new $$R_2$$) and subtract $$2R_1$$ from $$R_3$$ (new $$R_3$$): $$\begin{bmatrix} 1 & 2 & 1 \\ 0 & 3 & 3 \\ 0 & -3 & -3 \end{bmatrix}.$$ 3. Divide $$R_2$$ by 3: $$\begin{bmatrix} 1 & 2 & 1 \\ 0 & 1 & 1 \\ 0 & -3 & -3 \end{bmatrix}.$$ 4. Add $$3R_2$$ to $$R_3$$: $$\begin{bmatrix} 1 & 2 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{bmatrix}.$$ ### Step 3: Determine the rank The row-reduced matrix has 2 nonzero rows, so the rank of the matrix is 2. This means the dimension of $$W$$ is $$\dim(W) = 2$$. ### Step 4: Find the basis The basis consists of the linearly independent rows from the row-reduced matrix. Tracing back, we see that $$\mathbf{u}_1 = (1, 2, 1)$$ and $$\mathbf{u}_2 = (0, 1, 1)$$ form a basis for $$W$$. ### Final Answer: 1. **Dimension of $$W$$:** $$2$$. 2. **Basis of $$W$$:** $$\{(1, 2, 1), (0, 1, 1)\}$$. --- Do you need a detailed explanation of the row-reduction process? Here are related questions to deepen understanding: 1. Why is the row rank of a matrix equal to the dimension of the subspace spanned by its rows? 2. How can you verify if a set of vectors is linearly independent? 3. What is the geometric interpretation of the dimension of a subspace? 4. How does row reduction ensure the identification of a basis for the subspace? 5. Can the basis change depending on the choice of row-reduction steps? **Tip:** Always double-check for linear independence by solving for scalar relationships when choosing a basis!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Basis and Dimension
Row Reduction

Formulas

Rank of a Matrix via Row Reduction
Linear Independence Check

Theorems

Dimension Theorem for Vector Spaces

Suitable Grade Level

Undergraduate Mathematics or Advanced High School

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