Math Problem Statement
Find u = (u1, u2, u3) and v = (v1, v2, v3) such that the matrix A has the null space N(A) = span{(0, 2, 1, 1, 0), (-1, 2, 0, 2, 1)}.
Solution
The problem asks to find and such that the matrix
1 & u_1 & 0 & v_1 & 1 \\ 2 & u_2 & 2 & v_2 & 1 \\ 1 & u_3 & 1 & v_3 & 0 \end{bmatrix}$$ has a null space $$N(A) = \text{span} \left\{ (0, 2, 1, 1, 0), (-1, 2, 0, 2, 1) \right\}$$. ### Approach: We need to find the vectors $$u = (u_1, u_2, u_3)$$ and $$v = (v_1, v_2, v_3)$$ such that the given null space holds. #### Step 1: Null space condition If $$N(A)$$ is the null space, this means that for each vector in the null space, say $$x = (x_1, x_2, x_3, x_4, x_5)$$, it must satisfy $$A x = 0$$. Hence, we will set up the system for each of the null vectors: 1. $$(0, 2, 1, 1, 0)$$ 2. $$(-1, 2, 0, 2, 1)$$ #### Step 2: Apply each null vector For each null vector, compute $$A \cdot x = 0$$ to form equations that involve $$u_1, u_2, u_3$$ and $$v_1, v_2, v_3$$. Let me calculate and solve the system.The solution for $$u = (u_1, u_2, u_3)$$ and $$v = (v_1, v_2, v_3)$$ such that the matrix $$A$$ has the given null space is: $$u = (u_1, u_2, u_3) = \left( 0, -\frac{5}{2}, -\frac{3}{2} \right)$$ $$v = (v_1, v_2, v_3) = (0, 3, 2)$$ This ensures that the null space $$N(A) = \text{span} \{ (0, 2, 1, 1, 0), (-1, 2, 0, 2, 1) \}$$. Would you like further clarification or details? Here are 5 related questions to further explore this concept: 1. How do we interpret the null space of a matrix geometrically? 2. What is the rank-nullity theorem, and how does it apply to this matrix? 3. How do you determine the dimension of a matrix's null space? 4. Can a matrix have multiple null spaces, or is it unique? 5. What does it mean if a matrix has a trivial null space? **Tip:** In general, the null space of a matrix consists of all the vectors that map to the zero vector when multiplied by the matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrices
Null Space
Span
Formulas
Ax = 0
Null space of a matrix
Theorems
Null Space Theorem
Rank-Nullity Theorem
Suitable Grade Level
Undergraduate Level (Linear Algebra)