Math Problem Statement
Given the vectors (0,−1,2,1),(a,−1,0,1),(1,−1,b,0) calculate a and b so they generate a subspace of dimension 2. write each vector in a column
Solution
To ensure the vectors , , and span a subspace of dimension 2, they must be linearly dependent. This means that one vector can be expressed as a linear combination of the others.
Let's write the vectors as columns in a matrix:
0 & a & 1 \\ -1 & -1 & -1 \\ 2 & 0 & b \\ 1 & 1 & 0 \end{bmatrix}$$ To find conditions on $$a$$ and $$b$$ such that the dimension of the column space is 2, we can set the determinant of a $$2 \times 2$$ submatrix to zero. Let's consider the submatrix formed by the first and the second columns: $$\begin{bmatrix} 0 & a \\ -1 & -1 \\ 2 & 0 \\ 1 & 1 \end{bmatrix}$$ The determinant of a $$2 \times 2$$ matrix is: $$\text{det} = 0 \cdot (-1) - (-1) \cdot 2 = 2 \neq 0 \text{ (not dependent)}$$ Next, let's consider the submatrix formed by the first and the third columns: $$\begin{bmatrix} 0 & 1 \\ -1 & -1 \\ 2 & b \\ 1 & 0 \end{bmatrix}$$ For this determinant to be zero, we can calculate: $$\text{det} = 0 \cdot (-1) - (-1) \cdot 2 = 2$$ This condition doesn't help. Instead, we can set the third vector as a linear combination of the first two: $$(1, -1, b, 0) = x(0, -1, 2, 1) + y(a, -1, 0, 1)$$ From this, we can derive the equations: 1. $$x(0) + y(a) = 1$$ (1st component) 2. $$x(-1) + y(-1) = -1$$ (2nd component) 3. $$x(2) + y(0) = b$$ (3rd component) 4. $$x(1) + y(1) = 0$$ (4th component) From equation 1, $$ya = 1$$. From equation 4, $$x + y = 0$$ implies $$y = -x$$. Substituting $$y$$ into the first equation gives: $$-xa = 1 \quad \Rightarrow \quad a = -\frac{1}{x}$$ From equation 2: $$-x - y = -1 \quad \Rightarrow \quad -x + x = -1 \Rightarrow \quad x = 1 \implies y = -1$$ Now substituting $$x = 1$$ in $$a$$: $$a = -1$$ Finally, substituting $$x$$ and $$y$$ into equation 3: $$2(1) + 0(-1) = b \quad \Rightarrow \quad b = 2$$ Thus, the values of $$a$$ and $$b$$ that allow the vectors to generate a subspace of dimension 2 are: $$a = -1, \quad b = 2$$ Would you like further details or clarifications? Here are some related questions: 1. What conditions ensure linear independence among vectors? 2. How do you calculate the rank of a matrix? 3. Can you provide examples of linearly dependent and independent sets of vectors? 4. What is the significance of the dimension of a subspace in linear algebra? 5. How do these vectors behave in different dimensions? **Tip:** Always check the determinant of submatrices to assess linear dependence!Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Vector Spaces
Linear Dependence
Formulas
Determinant of a matrix
Linear combination of vectors
Theorems
Dimension theorem
Rank-nullity theorem
Suitable Grade Level
Grades 11-12