Math Problem Statement
Show that the given vector is in the span of S: a) 1 0 0 1 , S = 1 0 −1 1 , 0 1 0 1 , 1 1 0 0.
Solution
To show that the vector is in the span of the set , we need to determine if there exist scalars such that:
This gives the following system of equations:
c_1(1) + c_2(0) + c_3(1) &= 1 \quad \text{(from the first row)} \\ c_1(0) + c_2(1) + c_3(1) &= 0 \quad \text{(from the second row)} \\ c_1(-1) + c_2(0) + c_3(0) &= 0 \quad \text{(from the third row)} \\ c_1(1) + c_2(1) + c_3(0) &= 1 \quad \text{(from the fourth row)} \end{aligned}$$ We now solve this system of equations: ### Step 1: Simplify the system From the first equation: $$c_1 + c_3 = 1 \tag{1}$$ From the second equation: $$c_2 + c_3 = 0 \tag{2}$$ From the third equation: $$- c_1 = 0 \quad \Rightarrow \quad c_1 = 0 \tag{3}$$ From the fourth equation: $$c_1 + c_2 = 1 \quad \Rightarrow \quad 0 + c_2 = 1 \quad \Rightarrow \quad c_2 = 1 \tag{4}$$ ### Step 2: Solve for $$c_3$$ From equation (2), we have $$c_2 + c_3 = 0$$, so: $$1 + c_3 = 0 \quad \Rightarrow \quad c_3 = -1$$ ### Step 3: Conclusion The solution is $$c_1 = 0$$, $$c_2 = 1$$, and $$c_3 = -1$$. Thus, we can express $$\mathbf{v}$$ as a linear combination of the vectors in $$S$$: $$\mathbf{v} = 0 \begin{pmatrix} 1 \\ 0 \\ -1 \\ 1 \end{pmatrix} + 1 \begin{pmatrix} 0 \\ 1 \\ 0 \\ 1 \end{pmatrix} + (-1) \begin{pmatrix} 1 \\ 1 \\ 0 \\ 0 \end{pmatrix}$$ Therefore, the vector $$\mathbf{v} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix}$$ is in the span of $$S$$. Do you want more details on any step? Or further explanation? Here are 5 questions that build on this problem: 1. What does it mean for a vector to be in the span of a set of vectors? 2. Can a set of vectors span all of $$\mathbb{R}^n$$? Under what conditions? 3. How do you determine if a set of vectors is linearly independent? 4. What is the geometric interpretation of the span of two vectors in $$\mathbb{R}^2$$? 5. How would the solution change if we added more vectors to $$S$$? **Tip**: When solving systems of equations derived from vector spans, always check for consistency in each row to ensure the system has a solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Vector Spaces
Linear Combinations
Formulas
c1*v1 + c2*v2 + c3*v3 = v
Theorems
Span of a Set of Vectors
Suitable Grade Level
Undergraduate Level
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